Th,eIoumalot Gear' Manufacturin',g
SEPTEMBER/O'CTOBER 198B
Accurate gear inspection depends on retlat8d dataevaluation. Profit depends on speed. Maag CNC ~1 •• '" three, in the lab and on the shop floor. • Maag mechanical machines 'combine the speed and automatic checldng the reliability of a base circle disc design for profile and lead gear inspection. • Onlly Maag multi-axis electronic machines measure internal and external spur, helical and cluster gears with up to 15 sets of teeth, all in a single setup, with accuracy of .000040 in. • Maag's unique easy-to-use menu programming and software automatically bring the stylus to the moth surface. If you'd like to know how to spot your bad apples quicker, contact American Pfauter, 92,5Estes Ave., Elk 'Grove v.mage, IL 60007. Phone (312) 640-7500.
11111 !I.f" A .• 11-_..1••..:.,:., Iuou,n,· menea. r;nNllnnf'iii MIKRON brings you hob speeds upto 2,000 rpm and high rigidity with the A33/2 gear hObblng machine
The model A33/2 is a completely automatic gear hobbing machine with programmable controller. Users realize the full advantages of modem TiN coated hobs while hobbing gears of from 0.151" to 4" (4mm -100mm) outside diameter. The Mikron A33/2 is step-by-step expandable and is available in both automatic and manual load models. It can be fitted with an automatic deburring device and is also suited for skiving with tungsten carbide hobs because of the high rigidity achieved by its horizontal design.
write or phone for complete details on this flexible, expandable system and other Mikron hobblng. //£/fijj'" machines for gears up to 8" diameter. 71"7;ARCUT SALES,IINC.. &/ ~~~'61lndustrlal ParkDrive . - Farmington Hills, MI48024 313/474-8200 Telex 230-411 STARCUT SUPPLIED PRODUCTS AND SERVICES star Ma.cnl:neTOOlS / star cutting 'FOO'IS I Oolld'Star Coatlngls !Hurtn Macn:lneloois ! Mllkron IHob'blngMachines and Tools! TIN co.atlng Systems I stieber ,Clamping TOOlS
QIRCLE A-5 ON READER REPLYCARD EDI'I'OIUAL STAPP PUBUSHER • EDITOR·IN.QOEF Michael Goldstein ASSOCIATE PUBLISHER • MANAGING EDITOR Peg Short ASSOCIATE mrroR Nancy Bartels ART DIRECTOR Kimberly Zarley PROOUCDONIADVER11SING CONTENTS PAGE NO. Patricia Flam SALES co-oRDINATOR Jan LaBarge mrroRJAL ASSISTANTS Nancy Conn 'CROWNED SPUR GEARS: OPTIMAL GEOMErRY AND GENERATION 9 Mary Michelson Faydor L. Litvin, Jiao Zhang, Wei-Shing Chaing, Candace Rose University of Illinois at Chicagp, Chicago, lL IIANDAU. PIJBLISIIING STAFF John 1. Coy, Robert F. Handschuh, NASA Lewis Research Center, Cleveland, OH PRESIDENT Michael Goldstein VICE PRESIDENT CALCULATION OF OPTIM~ TOOTH FLANK CORRECTIONS Richard Goldstein fOR HELICAL GEARS 1.6 Manfred Week. Georg Mauer, Laboratory for Machine Tools VICE PRESlDENTIGEN. MeR. Peg Short and hldusttial Management, Technical University, Aachen, West Gennany ART CONSULTANT Marsha Goldstein ENHANCEU PRODUCT PERFORMANCE THROUGH eBN 'GRINDING 23 lANDAU. PUBUSHING Glenn Johnson and Ernest Rattennan, 1401 Lunt Avenue GE Superabrasives, Worthington, OH P.O. Box 1426 Elk Grove. D.. 60007 (312) 437-6604
EDITORIAl
TECHNICAL CALENDAR
IMTS ADV.ERTISER INDEX 8
BAiCK TO BASICS •.. 1NVOI.UTOMETRY 34 Harlan W. Van Cerpenand C. Kent Reece OUR COVER Van Cerpen-Reeee Engineering, Cedar Falls, LA Galileo's Escapement. This model, built 110m a of GaJileo 's. ~ constructed drawirW CLASSIFIED 46, in FIorenre. Italy, in 1883 and is now on display in the Science MusetII1I. London. Probably desiItned by GaJiIeo in the 1610's. the device is one of the earliest to me a pen- September/October. 1988 dtMum to ~ a docJnr;ork. It is an ap- plication of the principle. discovered by CaMeo. thai the period cf osdlation cfa pen- d&bn is the Si1I11e ~ cfthe size of the an: made by its ~ motion. It consists rfan esGlJ:Ie MbeeJ, pinion. pallets with 01.tch and the perxhium and its.suspension. (Photo courtesy of Sdence Museum, London, EtvJIand.) OperalOf Conlroll Panel for part Iload· inga.nd machine setup.
GraphiCS.- rlnter PIOtter delivers CNCstatus with copies ClfT. multl:-color hard monItor prOVides for one- copy 01 graphics status ai!d poSi- time entry of part and test data. tional display of p!int and tolerance mechanical system data. 'Mouse" per- 8.nd ONe control mits use of CAD functions. tE!(;hniques. Our Model 3000 ac Gear .Ana:ryzeris a third generation CNCgear inspection system in- corporating all of the comprehensive analytical tests and evaluation capabiilities of previous M & M systems, such as our Model 2000, but with these added capabilities: .' iDramat,ically improved speed and accuracy through new mechanical system design and advanced CNC control technology. • Computer hardware and applications software are modular to allow the user to buy onl,y the required, capability. This makes the 3000 ac adaptable to laboratory test,ing or producfien-llne inspection. • l:ntegrated Statistical, Process Control with local data base capability is an optlonal feature. . • Networking with MAPS compatibililty is available. • IRobotic interfacing for totally automatic :Ioadltestlunload operation can be incorporated.' . For more information or applications assistance, write or call: M & M Precision Systems, 300 Progress IRd.,West Carrollton, OH 45449,513/859·8273, TWX 810/450·2,626, FAX 513/859·445·2. M&M PRECISION SYSTEMS'
AN ACME-CLEVELAND COMPANY See the 3000 QC Cent.er at Boolh#6673, I!M!TS '88 CIROLS A-6 ON READER REPLYCARD THE 500 LBGORILLA IN THE WINDOW OF OPPORTUNITY
rMTS ISback. in [own, From Sept. 7 through Sept J 5. the largest Industrial exhibrtron In the Western Hemisphere will f,U one of the largest exhIbItion centers in the world, A show of this magnitude is a little like the 500 lb. gorilla In your dining room - hard to ignore. But like the gonlla in the dining room. IMTS raises certain ambrvrlenr feelings, True. I['S big and Important but what does It mean for our industry 7 The rest of the machine [001 business rends [0 dwarf the gearing Industry, and the ternpranon to Sitthe show out is strong. After all. there are other trade shows and conferences that relate more directly to our Interests, BeSidesIts bigness, what does IMTS have to offer the gear manufacturer, engineer or designer7 A lot. J .078 companies 11\1111 dIsplay the very latest machine tool products - including gear cutting and finishing equipment - from 27 nanons. There IS no better place to see the direction manuFactlJnng will take In the coming decades. Running concur- rently with the show is the Technology Conference, where some 200 industry experts from around the world Will cover every aspect of the lalest techniques In manufacturing in 48 separate half-day sessions. One whole session is devoted to modem gear making methods. and at least a half dozen other gear-relared topes Will be covered. That's a lot of information processes and rechnqoes available to lower your costs and and opportunity all in one place at one orne - information and improveyour quality, opporturrty we should nor Ignore. In a chess game. the winner ISthe player who can plan the America's industry has been through some tough times over most moves ahead, The same is true In the highly competitive. [he last decade. We have suffered from a slightly dulled com- global manufacturrng world In which we operate today petmve edge and. more critical In my opinion, a doJlarlforelgn- Education, Information and strategic planning are crucial to currency imbalance that no amount of cornpennve sharpness econormc survival. could overcome Now things have changed, Currency balances Don't let opportunities like IMTS slip by, Windows cI oppor- are much more favorable than they have been in years. and the tunity can close as quickly as (hey open. and the companies that harsh weeding out process has left Amencan Industry leaner. have not used their open time Wisely Will be shut out In the cold meaner and capable of faCing foreign competition Suddenly our products are In demand again. Exp0r!3 are growing and our rae- tones are busy. We have been able to come up for air In our struggle to survive the comperl[Jon. Bur we cannot be lulled Into complacency. This change for the better ISa window of opportunity. nor a permanent condi- tion. Exchange rates and economic Indicators are never fixed values. Politics. weather. scientific advances, an work to make the situation fluid, We have to make the oesr use of [he good times while we have them. Not only must we repair financial damage, but also reinvest some of our higher profits In our com- panies, in machines. people. education, systems and Informa- tion, In the years ahead. we should expect to face larger, better financed and even more stra[eglcally oriented competition. We must take advantage of trus present opportunity to make ourselves ready. lMTS is a good place to begin. Even If you're nor In the market for machinery today, trns isme place to seed what's out there. what the competlrion IS manufactUring. buying and will be using in the future ..A show like IMTS or the Gear Expo In Pittsburgh next year is the place to survey all the manufaaurjng options.
Sep'emberjOc'ober 1988 5 ------TECHNIC.LU C.LU4END1\.R
SEPTEMBER27-29. American Scxlecy cal Meeting'. Falfmon,t Ho,tel', N'ew MeridJan,.lndlanapolls, IN. The technical' for Metals 11th Annual' H'eat l\rea:tlng Orl'eans, LA. Presentations from U.S. conference will include papers on "Design Conference, McCormick Place, Chicago, and around the world on gear materials and Selection of Hobs.' "Selection of High- ll. Presentations on suqjects including heat and heat treatment design techniques. Speed Steel for Gear Cutting Tools", treating, statistical process control. new applications technology and the latest "Special Design Deburring EqUipment" energy applications, quenching and cool- technological developments. Contact: and other gear related subjects. Attendees ing improvements. For furtherinformadon. AGMA. 1500 King Street. Suite 201, Alex- will have an opportunity to meet with contact: ASM International, Metals Park, andria, VA 22314. 1703J 684..Q211. presenters to discuss papers and ask ques- OH44073.(216) 338-5151. tions on a one-on-one basis. Tuesdayeven- NOVEMBER 1'-3. SME,Gear.Processlng ing will feature a reception and tabletop ex- OCTOBER 10-12...AGMA Fall Technl- and Manufacturing ClinIc, Sheraton hibits from major manufacturers. For further information, call Dominic Ahearn at SME Headquarters. (313) 271-l 5()1Ox384.
NOVEMBER 5-10'. International ,Con- ference on Gearing. Zhengzhou', China'. ASME-GRI and several interna- tional gear organizations are sponsoring this meeting. For more information contact Inter-Gear '88 Secretariat. Zhengzhou Research Instittlte of Mechanical Engineer- ing, Zhongyuan ,Rd.Zhengzhou. Henan. China. Tel: 47102. Cable 3()()(). Telex 46033 HSTEC CN.
INOVEMBER8:.10. American Society for Metals Near Net Shape Manufac- turlng'Conferenc,e,. Hyatt Regency, Col- umbus. OH. Program will cover precision casting, powder metallurgy, design of dies and molds. forging technology and inspec- tion of precision parts. For further informa- tion contact: Technical Department Marketing, ASM International, Metals Park. OH 44073.
CALL FOR PAPERS - Tennessee Technological University for its list Inter- nat'l Applied Mechanical Sy.stems Design Conference', Mallch -19'-22. 1989, ,N'ashvllJe,TNI.Papers are invited on general mechanical systems sub.jectsin- cluding strength, fatigue life. kinematics, vibration, robotics, CAD/CAM and tribology. Deadline for first drafts is Oct. 1. 1988. For further information, contact: Dr. AVAIILA'BILITY, AFFORDABIUTY, DEPENDABILITY, Cemil Bagci. Dept. of Mech. Eng.. nu, . Cookeville. TN 38505. [61 S) 372-3265. THREE REASONS WHY AiEROSPACE SHOU LD BEYOUR DEALER! CALLFOR .PAPERSfor ASME 5th An- nual P·ower Tran,smlsslon & 'GeatJlng We specialize in gear cutting tools and gear machine accessories 01 many Conference. April 25-27, 1989', types inc.luding involute form CBN grinding wheels-hobs-dresser arms for Chicago,. !Il.Papers are Invited on emerg- Gleeson" grinders-cuNers for Gleason 104" -Gleason 114" and Revex" machines. Call us today with your gear tooling needs. CALL 312-323-0737 ing technologies for gears. couplings. belts, - --- - chains and other power transmission devices - gear geometry. noise, manufac- AEROSPACE, A. C. INC. ture. inspection, scoring. lubrication, materials, applications, efficiency.
m-.. ~UI!II,ca..:.1'4n .... ItIIdInIla,IItQuaW::lr'mEM!!I~~"&:i!IIIiIII!!iIIII!!I~~ G!IIIoII~WiE;i;ii!i!l~""~n"""""~ dynamics. For more .information, contact ~·''''''--''.ielPL Donald Borden. PO. Box 502. Elm Grove. CiRelli A-4 ON READER REPlV CARD WJ53122.
,6 Gear Technology
ADVERTISERS' INDEX
Booth #6263 Booth #6316 Booth #6226 • AMERICAN PFAUTER • HAMMOND MACHINERY MULTI-ARC 925 E. .EstesAve. 1600 Douglas Ave. SCIENTIFIC COATINGS Elk Grove, IL 6CXXJ7 Kalamazoo, MI 49007 261 E. Fifth St. (312) 640-7500 (616) 345-7151 St. Paul, MN 55101 16r 2) 290-6100 Booth # 6691 Booth #8016 .ielMA U.S.A. • JAMES ENGINEERING Booth #6192 50 r Southlake Blvd. 11707 McBean Dr. PEELCO MACHINE TOOLS, INC. Richmond, VA 23236 EI Monte, CA 9173.2 805 Albion Ave. (804) 794-9764 (818) 442-2898 Schaumburg, IL 60193-4522 (312) 35 1-2772 Booth #2102 Booth #6673 FHUSA •M & M PRECISION Booth #6136 American Inserted Hobs Inn 300 Progress Road PEREZ MACHINE T'OOl CO. r.o. Box 02-2837 West Carrollton, OH 45449 11 Ginger Ct. Brooklyn, NY 11202 (513! 859-8273 E. Amherst NY 14051 {718! 624-1607 (716) 688-6982 BoothH6027 Booth #61 19 MATRIX CHURCHILL CORP. Booth #6263
• 1 FRO MAG GmbH 5903 Harper Rd. • PF.AUTER/MAAG clo Roy Kemp and Associates Solon, OH 44139 Cutting Tools 38 W. 510 Lake (216) 248-7950 1351 Windsor Rd. Charlotte Ct. Loves Park, IL 6113.2 St. Charles, IL 60175 Booth #6392 ISI5J 877-8900 [3 12) 584-4488 • MITSUBISHI HEAVY INDUSTRIES AMERICA,. INC. Booth #8102 Booth #6301 873 Supreme Dr. • SU-AMERICA • THE GLEASO.N WORKS Bensenville, IL 60 106 8775 Capital Ave. ICXXJUniversity Ave. (312) 8604220 Oak Park, M148137 Rochester, NY 14692 13r 3) 548-71 77 (716) 473-ICXXJ Booth #8416 MOREY MACHINERY .1 • For additional information see Booth #6691 22 I Cottage St. ad elsewhere in this issue I. GMI~MUTSCHLER Middleton, NY 10940 19 W. Chicago Ave. (914J 343-1851 Hinsdale, IL 60521 (312) 986-1858
8 Gear technology Crowned Spur 'Gears: Optimal Geometry and Generation
Fay-' d .or" L Litvin-' . , _laOJ" Zhang,· W· .eJ- . S"L!-(h'" -,rung ,·amg, U" ,mvers!"ty ,'.- 0'- . f· ,_-.015,,-.-Olin" , Chi'ca.go John J. Coy, Robert ,F. Handschuh, NASA Lev.ris Researdll Genter, Cleveland, OH
Abstra,ct: a) The authors have developed a method to. synthesize the pinion b) crowned surface that provides a localized bearing contact and a favorable type of transmission error for misaligned gears. A method for generation of the pinion crowned surface bya surface of revolu- tion (it slightlydeviates from a regular cone surface) is proposed. Tooth Contact Analysis (TCA) programs for simulation of meshing and bearing oontact for misaligned spur gears with the crowned pin- ion have been developed. A computer graphic program for·the display of the pinion crowned tooth surface in 3·D space has also been developed. c) d) Introduction Involute spur gears are very sensitive to gear misalignment Misalignment will cause the shiH of the bearing contact toward the edge of the gear tooth surfaces and transmission errors that increase gear noise. Many efforts have been made t,oimprove the bearing contact of misaligned spur gears by
crowning the pinion tooth surface, WiJdhaber(1) has pro- Fig. 1 posed various methods of crowning that can be achieved in the process of gear generation. Maag engineers have used dresses only half the problem. The transmission errors of crowning for making longitudinal corrections [Fig.1a); modi- misaligned spur gears are a main source of gear noise ..Ac- fying involute tooth profile unifonnly across the face width cording to the open literature, 'the influence of gear misalign- (Fig. Ib); combining 'these two functions in fig. Ie and per- ment on transmission error has not been inv,estigated.Maag's fonning topological modification (Fig. Id) that can prov:ide method. of topological modification does not determine the any deviation of the crowned tooth surface from a regular relationship between the surface deviations and the transmis- involute surface. (2) sion errors, Also, the optimal geometry for the pi.njon The main purpose of these methods for cr-owningis to im- crowned surface has not been proposed. prove the bearing contact of the misaligned gears, which ad- The contents of this article cover the solutions to the
AUTHORS:
DR. FAYDOR L. lITVIN.is Professor of Mechanical Engineering DR.. JOHN ,. COY is Branch Chie!, Rotosystems Technol~gy, Pro- at the University of Illinois at Chicago. He is the author of severa.1 pulsion Systems Division, NASA Lewis Research Center, C1eue.1Dnd, books and many papers on gears and holds several patents ..Dr. Litvin OH. He has been~esearching helicopt.er d.rive trains since 1974. Dr .. has served as a consultant to a number of industrial corpor,UioTl5. CO'lIearned.his undergradl.lllte and grad'lJQtedegT'ees from the' U'niver- He is chairman of the ASME subcommittee on Gear Geomefry and sityof Cincinnati. He has served on the faculty of the Rose' Hillman a member of the ASME Pouer Transmission and Gearing Commit- Institute of T,echnology and the Ul'liveTSity of Cincinnati. He has tee. He has won a number of professioMI awards, including three published over 65 technical papers and is ,a consultant to ,industry NASA Tech-Brief ,awards for published papers. and other government ,agencies..Dr. Coy is a member of A SME and .a licensed professional engineer in the State of Ohio,. MO .ZHANC is a doctoral candidate in the Mechanical Engineer- ing Department at the University of Illinois ,at Chicago. He is cur- MR. ROBERT F. HANDSCHUH works for the u.s. Army rently researching computer modelling and simulation for gearing AVSCOM Propulsion Directorate, NASA Lewis Research Center, theorems and design. Clevela.nd, OH, in the area of gear geometry, gear manufacturing and drive train performance. He did ,undergraduate work in WEI~SHINC CHAING is a doctoral cllndidate at the University mechanl'cal engineering at Cleveland State University and enmed' 1m of Illinois at Chicago in the Mechanical Engineering Department. His MS in mechanical engineering from the University of Toledo. Mr. current field of research is computer"aided simulation and design for Handschuhis a member of ASME and the National Society of Pro- gearing. fessional Engineers. ______~ September/October 19,88 9 J followmg problems: opdmalgeornetry ofa pinion crowned aq;2 tooth surface; new method of crowning that is based on ap- plication of a tool. provided with a surface of revolution (the a) tool surface is slightly deviated from a cone surface); the development of TCA {Tooth Contact Analysis) programs for ~ /1 /' the determination of transmission errors .for misaligned gears ~I and their bearing contact and a computer graphic program ~v V for the display of the crowned surface and bearing contact in 3-D spaoe ..The first method of generation that provides the desired optimal pinion 'tooth geometry is basedon the application of a computer controlled machine with five degrees of freedom ..The second method for generation needs b) only a new tool. shape and is based on application of the ex- isting equipment. The development of the optimal geometry of a pinion
crowned gear tooth surface is based on the following ¢I considerations. Misaligned spur gears with a pini.on crowned tooth sur- face can provide transmission errors .6.412(411)of two types, shown in. Figs. 2a and 2b,. respectively. The transmission Fig. 2 errors are determined with the equation as and the gears are misaligned; 41,0= N1q,1 is the theoreticsl aq,2 = ~(q,1) - Nl. rbt (1) 2 N2 N2 relation between the angles of rotation of the gears in the ideal Here: Nl and Nz are the numbers of gear Iteeth; q,,1 and q,2 case where the gears are not crowned, not misaligned and aile the angles of gear rotation; q"z(q,t) is the function that the transmission errors do not exist. Type 1 transmission relates the angles of rotation of gears Hthe pinion is crowned errors are not acceptable because the change of tooth meshing is accompanied by an interruption or interference of tooth surfaces ..Type 2 transmission error is preferable if the level of 'error does not extend the prescribed limit. 1988 At first glance crowning should be directed toward pro- viding an exact involute shape in 'the middle cross section (Fig. Fall Technical 3). In realjty, this ,type of crowning is not acceptable because Meeting the misaligned gears will transform rotation with transmis- sion errors of type 1(Fig. 2a), but not of type 2 {Fig. 2b). New Orleans • October 10 - 12 For this reason the authors have synthesized a specific pinion crowned tooth surface .. Sucha pinion,. while in mesh wi.tn Jazz Up Your Technology a. gear that has a regular involute surface, is able to provide transfonnation of rotation with a parabolic 'type of transmis- sion error function. This 'type of function of errors is syn- Topics Include: thesized for ideal gears that do not have any misalignment. Then, the tendency to provide parabolic transmission errors Practical Carburizing Specifications can be extended Ito misaligned gears and the discontinuance Case Hardening in Large Gear Units Calculation of \lVormgear Load of meshing can also be avoided. Note that the proposed Miner's Rule .Applied to Industrial Drives method of synthesis provides a shape in the middlecross sec- Special Problems in Gear Tooth Couplings tion. of the tooth that deviates in a certain way from the in- IDevelopment of a Noise Test Facility volute curve that is shown in Fig. 3..The longitudinal devia- lDynamic Behavior of Epicyclic Gears tion from a straight line is not related to the transmission er- A New Approach for the 'Derivation of rors, but to the desired dimensions of the instantaneous con- lDynamic Load tact ellipse for the gear tooth surfaces. The proposed pinion Load Evaluation of CBN Finished Gears tooth surfacecan be generated by a plane chosen as the tool .A New Process for Finishing Bevels surface. The rnotionsof 'the plane with respect to the pinion Reduction of Transmission Error must be controlled by a comput.er,and the machine represents For Reglstratlen Informatlon Conteet: an automatic system with five degrees of freedom (first AGMA Headquarters method for generation). (703) 684-0211 The second method of pinion crowning is based onap- plication of a surface of revolution that slightly deviates from a regular tool conical surface (Fig. 4). Such a tool can be usedl CIRCLE A-24 ON READER REPlY CA!RD as a grinding wheel. or asa shaver. The motions of the tool
10 Gear Technology and the gear being generated are related sim~larly to the mo- tions of a rack-cutter and the gear .. (See Section 3.) A tool witha regular conical surface can generate a pini.on crown- ed surface whose middle cross section represents an involute curve (Fig. 3). However, this type of pinion crowned sur- face is not desirable because the misaligned gea.rs provide Type 1 transmission errors (Fig. 2a). Therefore, a tool with a surface of revo1ution instead of a conical surface is used. The evaluation of the bearing contact and transmission errors for the misaligned gears, as well as the investigation of the influence of errors of gear assembly, needs the applica- Involute shape tion of TCA programs. The programs are based on the following algorithm: (a) The contacting gear tooth suriacesare represented in a fixed coordinate system, Sf, thai! is rigidly connected to, 'the gear housing (Fig.. 5). (b) The continuous tangency of gear tooth surfaces is pro- Hg.3 vided if the position vectors and surface unit normals for the contacting surfaces coincide at the contact point at any instant. Then we are able to determine the path. of contact on the gear tooth surfaces and the relations between. the angles of rotation of the output and input gears. Knowing function 4>2(4)1) we can determine the deviations of this fundion from the prescribed linear function; i.e, the transmission errors. (c) Due to the elasticity of the gear tooth surfaces, the sur- face contact is spread over an elliptical area ..The dimensions
Fig .. 4 ::a: Q,
~ ~ £1:' 'c:::i iQ.,' c:r: '.GFUNOING • BEVEL Q, .SKIW,IING 01 SPIRAL r ....•TOOTH RAIDIUS • PINIION ...., • BRUSH liNG ·SPUA • ZERO SI:T·UP " HELICAL ~ • WHEEL. CHANGIE '. FUNGGEARS ... , 5SECONDS I...... ,
I~I 1817 -18th Ave. 1/ ILLj RocKford, IL 61 "04 815~398~'010 FAX.815-398-1047 11 ICC .DEALERS WEL.OOMEI
Fig. 5, CiIRCLE A~21 ON READER RERlY CARD
September/October 1988 11 and orientation of the instantaneous contact ellipse depend on the principal curvatures and the principal directions of the contacting tooth surfaces. The bearing contact is deter- mined by the developed TCA program as the set of the con- tact ellipses that move over the contacting surfaces in the pre- cess of motion. a)
Synthesis of Pinion Crowned Tooth Surface Consider that the gear is provided with a regular involute surface. The pinion will be provided with a crowned surface. The shape of this surface in the middle cross section is syn- thesized on the basis of the foHowing considerations: Two o shapes - a regular involute curve of gear 2 and the to-be- determined shape of the pinion tooth surface - are in mesh in the middle cross section. Although the gears are not b) misaligned,their shapes, being in mesh, must transform rota- tion with the function (Fig. 6a).
rp2{,rpl) ;;;;;; rp1 ~l + .a.rp2(rpl) = N2 Middle-cross arp2 sect.ion shape 1 Here: (2) a (3)
.tlr.b2(rpl) is a parabolic function that must satisfy the follow~ tig.6 ingequation :.
11"
N1 (b - arp2)dr.bl = 0 (4) 11" 1 11l_GARA------GEAIR CDR~ 1 -Nt 941 MILITARY RD. Here 211"is the angular distance between two pinion .r-r-I'"j!,s Nl ~ BUFFALO, NY 14217 neighboring teeth. Equation 4: states that the arithmetic OVer 30 VealS Continuous Service to Industry average of the transmission error tlrp2(·tPl) over the GEAR GRINDING interval (- 11', _!J is equal to zero. N1 NI SPECIALISTS After some transformation we obtain - MFG,..ALL TYPES INCLUDING tPZ(rpl) = rpl NN1 + d [113 - (Nl)2r.b~]' (5) .• Spur • Complete Machining and 2 7T • Helical Grinding Capability ··Internall • Includes: O.D.alld: I.D. The magnitude of d represents the level of transmission er- • Pump Gears Grinding', Gear Honing ror. Using methods of -synthesis of planar gears(3) we may • Splines and Pulleys w/Crowning. Broaching, .• Serrations Keysealing. fuming and determine the sought-for shape of the pinion middle cross • Sprock·els and ,Ratchet Milling, Tooth Chamler- section. fype'Gears ing,and Rounding The longitudinal shape of the pinion crowned tooth sur- • Hobbing up to 24" in iDiameter face may be determined from the requirements of the con- tact ellipse. The authors propose representing the pinion CALL NOW IFOR A QUOT,E crowned tooth surface as a surface of revolution that can be • Supplied complete to print generated by rotation about its axis a-a (Fig. 6b). • Finishing operations on your blanks The advantage of the proposed geometry of the crowned • Grind teeth only pinion tooth surface is that ~thegears, while misaligned, have FAX. (716)874--9003 • IPHON,E (71!6) 874-3131 a parabolic type of transmission error, and the discontinuance of meshing can be avoided. An example is given below to demonstrate the results that
12 Sear Technology can be obtained through use of the proposed crowning or as a shaver. The opposite sides of t.he pinion tooth are method. generated separately. Example 1 The described process of the crnwning of the pinion by Given: numbers of teeth: N) = 20, N2 = 40; diametral a regelarcone provides an involute shape for the pinion tooth pitch, P = 10 1; pressure angle t c = 20°. The pinion surface in its middle section. Th crowned pinion and the in involute gear. if they are not misaligned. can transform rota- tooth surface has been designed as a crowned surface with tion without transmission errors, and their bearing c,ontact a parabolic tr,ansmission error maximum d = 2 arc seconds can be localized. However. the misaligned gears will in the aligned condition. The developed TeA program has transform rotation with Type 1 transmission errors (Fig. 231). been applied for the evaluation of transmission errors for the To avoid the disoontinuance of tooth surfaces that occurs at following misalignments: the change of teeth in meshing. a surface of revolution must (i) The change of the center distance is 4C = 1%. The be used instead of a. cone surface, This surface slightly c deviates tram a regular cone surface and its application for gear axes are not parallel, but crossed, and the screw angle crowning provides Type 2 transmission errors (Fig. 2b). Also. is five arc minutes. The function of transmission errors caused by the misalignments mentioned above is of a parabolic type. and it is represented in. Table L The maximal value of transmission errors is 1.2 arc seconds. (iii) The gear axes are not parallel, but intersected. and form
the angJe (l = 5 arc minutes. The function of transmission errors is of a. parabolic type. and the maximal! value of transmission errors is 2.0 arc seconds (Table 2).
Generation of Pinion Crowned. Tooth Surface by a Tool With a Surfaoe of Revolution Fig. 7 shows the installment of a tool with a regular cone surface. The cone surface is tangent toa plan.e whi.ch is the
surface of a rack cutter. We can imagine that the two tools - a Fig. 7 cone and a, rack cut ter - being .rigidlyconnected, generate a. crowned pinion surface and a regular gear involute surface, r,espectively. In the process of generation the rack cutter and the cone perlorma translational motion, while the pinion and the gear rotate about their axes (Fig. 8). The rotation FROM,~ of the con about i1s axis, C-C, is not related to other mo- IGEAR DEBURRIING MACHINE tions thae have to be provided for the tooth surface genera- tion. The angular velocity in the rotational motion of the cone ., Deburrs any contour, even hellcallsplrals depends on the desired velocity of cutting ..The tool for the .1 Quick set· UP', easy to change crowning of the pinion can be designed as a grinding wheel • Semil·automatlc, h,lgh production
Table 1 function '0£ Transmission Errors
Table .2 Send for a free, catalog todayJ See us at IMrS '88 Booth 6316.
~;?f~,-.-_. ,.-.-~-"'i:.l 1600 IDo\!glu, KalIIIIUOO. MI >40007 616 f 345-7151
CIRCLEA-28 ON IREADER IREPLY CARD Septem'ber/October 1988 113 knowing the topology of the pinion crowned surface that is with the following results. The gear axes are not parallel, but generated by the surface of revolution, we can generate it crossed, and the screw angle is a = lOarc minutes. The bya plane. The conditions of gear meshing and their bear- change of the center distance is .8.c = 1 %. The function ingcontact have been simulated by the TCA program that c has been developed by the authors. An example is shown of transmissionerror is of a parabolic type, and the max- below to demonstrate the concepts developed in this article. imal value, 0.35 arc seconds (Table 3) .. The gear axes are intersected and form an angle, a = 10 Example 2 arc minutes. The function of transmission error is of a The input is the same as in Example 1. The pinion tooth parabolic type and its max ..imal value is 0.34 arc seconds surface is crowned by a surface of revolution with the follow- (Table4). ing parameters (Fig. 9): e = 20°, R = 500 ", The misalign- ment of gears has been simulated and the transmission er- Table 3 rors have been evaluated by the developed TCA program
Table 4
Fig. 8 If you sell to the Gear Industry This is your Show!
o AGMA is building on the success of Gear Expo '87, which attracted 1200 visitors from around the globe.
D Gear Expo is held in conjunction with AGMA's FaH Technical Meeting, ensuring a targeted audience dedicated to the gear industry.
D Our audience packs purchasing power! 80% of our attendees can influence purchasing decislons or have decisionmaking authority.
Q We a.re already 80% committed! Last year's exhibitors are returning and asking for even more space! Gear Expo '89 Your competition will be there! •.. - November 6 - 8, 198ge Pittsburgh, PA ~- For more Information contact Wendy Peidl at AGMA. (703) 684-0211 _...... •
CIRCLE A-25 ON READERREPLYCARD
1'4 Gear Technology " TeA programs to simulate the meshing and bearing 'con- tact of misaligned spur gears with th~ crowned pinien and to investigate the influence .of misalignment on the transmission errors .. • A computergraph:ic program to display in 3-D space the pinion crowned tooth surface and the Jecation and orien- tation of the contact ellipses.
References 1. WILDHABER,E. 1962, Method and Ma.chine for Pr,oducing Crowned Teeth. USA Patent 3.046,844. 2.. Maag Information 18, Topological MDdjfica~jon. Zurich. 3.. lITVIN, F.L. 1968, Theory of Gearing, 2nd ed, Moscow (.in Russian). The English version is in press, NASA Publications.
Reprinted with permission of tire American Gew Mmll~fQclUFflrs Association. The opinions, statements Qlul conclusiom presented in this article are thoSl' of the authors and in noway represent the position or opinion of the AMERICAN GEAR MANUFACTURERS ASSOClA'HON.
Fig. 10
Display o.f Analytical. Resullts iby Computer Graphics A computer graphics program was developed to represent Drewco Holblbing IFIil:l(tur,esmaintain the in 3-D space the pinion crowned tooth surface and the mo- r~quilre_C!..~clos,e .~elationshilP betw.een the looatllng surface ,andltihe falce of tion of the contact ellipse in the process of meshing .. the 'g,ealr. Tlhle rUlgged precjsten The graphics program is based on the analytical. solutions construction rnatntalns uhe close hold that have been obtained from the TCA program. It produces to face retatlonshlp whether locating the intricate high resolution picture on a graphic terminal and on smooth diameters or spline teetli. laser printer. Fig. 10 shows the results of the computer graphic The expanding a..rbee w.iII. ~ff,ectiyely hotd a gear fllankfor flnlSh, turnmg, program that represents the pinion crowned surface and the gear cuning. glearfinishiin,g, and giea:r location and orientation of the contact ellipses. Inspection operations. -
Conclusion Center Mandrell •.Adjullable HoldeN • checkinGI Flfiu .... Centering Chuc:kl • 80rln1l118.r'CIUICkl • Splndlel Chuclu The authors devdoped: " A method for the synthesis .of a pinion crownedsurface ,that provides localized and a limited !DREW'C'D a beariagcontact CO.R:P.QR:A.TION - - level of transmission error ofa parabolic type. This ap- Precision Workholdlngl and Tool Holding Devices proach provides favorable conditions of meshing and 3745 NichOlson Road • P.O. Box 127 contact for misaligned Spill gears. Franksville, WI 53126 •A method for generation of a pinion crowned tooth sur- Telephone (414) 886-5'050 face by a surface of revolution that slightly deviates from IFax iNumiber (414)886~5872 a regular cone surface. This method can be applied fer crowning by grinding and shaving. OIOCIJE A-14 ON IREADER' REPLY CARD
September/OCtober 1988 15, Prof. Dr. -Ing. Manfred Week Dipl, -Ing. Georg Mauer laboratory for Machine Tools and Industrial Management (WZL) Technical University, Aachen,West Germany.
Abstract: Strider reqWre.ments are being imposed on types 01 rootnr lank correct ions F'lg. 1- (upper left) Objectives of heavy duty gears in terms of running behavior tooth flank corrections: and load carrying capacity.Purthermere, the • reduction of pressure peaks • decrease ofdisplacement senstvity expanding teclmiquesin the field of gear II p!o'fLJe: cor"Ffc.UDf'I ., lmal tytllnti c:arrCUIM • diminishing of engagement shocks mjU\ufacturing are demanding 'the develop- • lessening of parametric excitation ment of suitable designing methods for tooth flank corrections. In order to calrulate these corrections, the spatial stress and deformation ji§ ~ state in the mesh has to be determined. This c, tit Df.fj] till article reports on the further development of a () ~ _aJlel 1Q JI.. !; 11'1the" illlC:MlWI 1:oruc.upn finite element calculation method into an op- timizing and designing system for flank OOITec- tions on spur and helical gears. ro··· c.- tlJ:." Fig. 2- (below) Basic principles of Main Influences on the Bearing Capacity contact analysis of loaded gears .. and Run_ning Behavior of Gears
The load carrying behavior of gears is calculation at the calculation of ttle preset geOletrl.C strongly influenced by local stress concen- engagement cona it Ions gear Dllalll.11 ties tooth aevlaUoos trations inthe tooth root and by Hertzian alstortlona,l Ile!iavlour pressure peaks in the tooth flanks pro- of ttle llesh duced by geometric deviations associated with manufacturing, assembly and defor- mation processes, The dynam:iceffects diS tor tIona I. .tlehav.IOOI' ofllheel. si1aft !J1d within. the mesh areessentially determined DeBrJngS by the engagement shock, the parametric excitation and also by the deviant tooth .'alscrete cootact oorms of • utrh ,tt. _Uh defOr.lllltlon Indices cootact III steoces s for O!lffelent roiling [JOsltloos ~ for discrete [JOInts discrete [JOInts geometry, (1) The engagement shock results from a displaced starting point ofengagement soiu; Ion of the contllCt-Drool.eII In !)ear eng&gell!ntsfor dtscrete oolnts of different roll.1119 pas I tloos due to rotational deviations or pitch errors n' F + 5 • ,Srj91!1' marolna·1 cOOClltlons : I:F I • ftot ' Sn91a ' I." censt, within the gear system. This transferred start of engagement is located outside the plane of action. Here deviations occur in alstr1butlon of 10<10 af'll1 pressure the value and direction of the normal tota 1course llesh st cttrrness "1 velocity 'components of the contacting tooth flanks; thus, vectorial difference contact oattern
16 Gear feohnology These precision Bryant CNC grinders, the finest The Lectraform grinders can handle parts available anywhere, are ideally suited for either from 0.050" to 24· 00 and are available in short-run or high production performance. Our two standard slide arrangements. Unlike other Teachable® III CNC unit permits us to precision manufacturers a compound slide is out of the grind parts so varied in size and shape (even question. The closer that big grinding wheel is to out-at-round intentionally) that you wouldn't the bearing support, and the Lectraform can use believe the same type of machine could do it all. larger wheels than other makes, the greater the They perform contour grinding - both rotary and precision - and that's the name of the game. linear, vector grinding, conjugate grinding, and Tell us what you have to grind and we'll show you simultaneously or sequentially grind multiple how to do it - precisely. Bryant Grinder Corporation, surfaces in a single set-up to provide ultimate Springfield, Vermont 05156, (802) 885-5161. precision and concentricity. Our time-tested Commonponeni® construction permits the use of identical slides, drives, ball screws, controllers, dressers, etc. on either machine. That's one reason why they are so flexible, precise - and affordable. Commonality of parts - bet your bottom line will love it! We offer slide resolutions as fine as 0.000001" and when it comes 10 sizing capability, we've got it. The Lectraline machines offer the world's widest capacity range- 0.040" to 88" in diameter with strokes up to 7 feet. They come in over a half-dozen standard configurations to suit your exact requirements. BRYANT GRINDER CO'R,PORATIOiN
CIRCLE A~7 ON IREADER IREPlY CARD brings about an unwanted impact speed decrees 01freedom" and shockacnvation.V' dJeclgng of number 01describing 'Quantity to QPhmlze : restrjgtipns " The parametric excitation is a conse- vanabiesfOi the tooltl Hank correction quence of t.hechanging number of mating flank cerrecnon .' contact rano teeth and the contact line movement over • rOiling conditions the tooth flank during engagement. As a
• contac1 conditions and result, a periodic modulation of the mesh , ~tqyanti1ies" stiffness follows and induces rotational • ,engagemefll conditions 100111contact analysis correctional motions In 'the mesh of loaded gears tly the deviations and activates unwanted vibra- • ccmpliarces at manulacMing stage • outer torque methodol6nite elements tions, even when the outer load f.or the .' deviations gear is constant. (3)
Geometry of Bank Corrections cnlerions lorHfIOk corrections' oPhmjljDQ, me for Helical Gears loom !lank correctIOn; • bene[icial • distritJution or load The flank geometry of helical gears can • varlafional calculation load characteristics and pressure to be modified in various ways with respect • low dlsplaoement .' modification 01the the mating, area to manufacturing requirements and the sensivity oorrec1ion topography • rota.tional deviations differential improvements on the bearing • low dynamIC • checking the of loaded gears excitation convergence '1:lel1aviour capacity and dynamic behavior, Fig, 1 gives an overview of the principle correc- I ~ © tionforms. Hanl. corrections in the direction of Fig. 3-Optimization of tooth Hank corrections. tooth depth as shown in Fig. la are simply carried out by involute tip or root reliefs. In this case, the part of the flank to be taken back consists of a corrected involute profile, which is defined by a. modified base circle diameter and by the intersec- tion point with the uncorrected involute ... ~o curve, This is achieved by tools having a ...)6 basic rack system with altered profile <: mm 0- angles or by using a modified working ....u Q 20 pitch diameter at generation. Another form of profile correction which has a s~ 15 ...C1 smooth transition to the original, uncor- 0 11) rected involute curve is manufactured by .c .... :::=::::"0-- 0---- tools with crowned basic rack profiles . ..<: 5 ~~OfPlnlon~5 _ QJ o~ 1 Profile corrections are mostly used to G 1 I - ,~ 1 1 ---r---~- decrease the engagement shock and the in- o 10 20 30 40 50 mm 70 face ,wIdth gear data: AUfHORS:
lengtl1 of PROF. DR.-INC .. MANFRED WECK path lin ; 7 IIIiII correc t I,on of contact holds the university chairoi Machine Tools at value Z, = 27 the Technical University Aachen. West Ger- 40 l2 ; 28 many. He is also director of the Laboratory for kim x, -0,006 Machine Tools and lndustrial Management 20 (WZL) and manager of the machine tool de- x2 -0.031 partment of the Fraunhoier 1I15titutefor Pro- o fa" '.55 duction Techno.logy ([PT) in Aachen. Prof. Et3 = 1.00 Wecks professional interests are in the areas of machine testing and evaluation, machine Il:n = 20· design and calcula.tion, ,development of auto- fj = li6· matic control and monitoring systems, auto- mation of production and ha.ndling ,and gear b ; 78,8 !11111 cutti.ng machines and gears. a = 200 .. '1 = 4820 NIl OWL. -ING. GEORG MAUER received his degree in mechanical engineering at ,the Tech- nical University of Aachen and is curren.tly wo rking as a scieniiiic assistant in the gear Fig. 4-0ptimized three-dimensional correction. with basic gear data. research group of WZL. volved strain and noise, \1lIO\~ ._ Mtrl1tll1!lft preuurn longitudinal. flank corrections in the 0<1 roiling nilS Of! tllo_1IrH direction of the fac,e width (Fig, tb) are used to attain. a, low displacement sen- sitivityand to avoidstl'ain peaks ocruring due to displaced positions of the gear axis ('twisting, tilt) or helix deviations of the Hanks, For hehcsl gears, a correction running parallel to the lines ofeontact is advan- tageous. (See Fig. Ie.) The intersection line _ The correction fonns described above o • • WII.·ID' II are to a eertain extent funited in optimiz- rltYDLuUDftlID.llfItIlt "1 ing the rumnngbehavior and load carry- ir!g capacity. because the flank corrections ! -~I 11.10 do not meet the gear geometry and mating I I IIi 1.05 conditions adequately, These conditions J can be met by using three dimensional 1 111.00 ml:;:;...~=::;:::;=i 1 2 , • ·S 6 1 , 9' ID o • I U/IIln'IO" n (oHen called topologjc(4l) flank correc- ,,~ ."tl' "'D '_luU .... 0••1...11.'" Hon~ (See F.ig. ld.), which are character- ~ ized by variable correction forms in both .., Ii '1 _ iln ~ 2Oi! ., • 71· ", - -.ID ~a· '1.55 directions of tooth depth and face b •• 11,' !!!! ~ • 16' '1 • ,.OJI 'II' 1.00 " • 1iI2O: .. width,(Sl There are, however, various marginaiconditions which have to be taken into, consideration. these being the 11"18. ·6 - Periodic courses of the total mesh stiffness. continuity of dilierentials.minimumcon- tact ratio, rolling conditions and manufac- ~ 7i turing processes. (l) I_M_ ~ ,...... ::: .,'U. ~r .. t .. Optimizing Method Based m, Finite ~ - Element Ca!letdatwons 1 5 ffc,'''_ •. ....- .. --- ...... ~ Cattltt .. To obtain. the required starting position s: 10 I to optimize flankcorrections. one has to e 100 '1"2 • l'S," , I I return to the basic principles of contact 11/12 • O.5OQI1l~26I J e • J«IJ I!I!!! ~ I~ analysis ,of loaded gears illustrated in ~l/a02 • 599._ .... 1~1'.~· 011', '·20' j L ,,- c~~tld . fi fig. 2. The continuous, spatial load defor- 199th'l - COUKtll7n 'ot '1 • .a UII : ~ 1\..:)1' I mation problem is replaced by a system of 100000!ti.KIUMll 'Md reUd's .iI'./"" " 0 I '"-~ discrete contact points. defined by a 1500 ~ r..1nttoa matrix with deformation indicies !a-I, a --~,-" , UJOO load vector {F}and a vector containing .' , Iii 750 / . ~-- contact distances [s], which describes the -,"'lCtld \ ~ 500 geometric flank deviations or corrections. 110 III 2SO The equation system is then set up for dif- i I ferent rolling positions and is solved by considering Ithe marginal conditions con- c,eming the 'totalload and the rigid body .Fis. 7 - Smiples of geometrically simple corm:tion forms. September/Octo'ber1988 J 9 displacement. (6) The results provide in- the mesh and analytical models of shafts as described above, presents all the criteria formation on the load and pressure and bearings. necessary to assess a particular correction distribution on the field of action and the This contact analysis is the central start- with regard to the various objectives. The shape of the rotational deviation curves, ing point to optimize tooth flank correc- mathematical variation provides informa- which is influenced by the alternating tions ..(See Fig. 3.) By varying thecorrec- tion regarding the trends and dependent mesh stiffness. It becomes clear, therefore, tion topography, a beneficial load and variables of the topography and the target f.haf Ioad distribution and mesh stiffness pressure distribution on the flanks, a I'ow objectives, so that the correction can be are direct functions of the correction to- displacement sensitivity and a lower improved in a stepwise fashion. This pography. The pliability indicies tal are dynamic activation can be achieved. numeric process has to be repeated several calculated by finite element structures of The con tad and deformation analysis, times after checking that all the restricting marginal conditions are met. Other research developments at WZl concern the simulation of grinding proc- esses for corrected tooth flanks, whereby correction movements and contact condi- tions are analyzed ..Control data for NC grinding machines are also generatedPJ Calculated Examples The following section contains the calculation results of two flank corrections for typical helical gears in industrial. use .. Fig. 4 shows an optimized three-dimen- sional correction with the basic gear data .. The correction values are drawn up on the plane of action with a maximum relief of approximately40p.m. In the middle of the engagement area an uncoerected region ensuring a contact ratio greater than 1.0 can be seen. f1) The correction topography is observed to run parallel to the original involute flank at the approach contact, which is favorable fQr lowerexcuation produced by engagement shocks, The calculated tooth loads for a full cycle of meshing contact are illustrated in Fig. 5. A lower force level at the start of engagement and a shallower Ioadreeep- tion gradient for the corrected gear ean be seen dearly. The pressure distribution on the plane of action depicts lower strains for the corrected gear, and the pressure peaks have been successfully reduced where the contact lines end at the root ·0£ one of the mating teeth. InFig. 6 the periodic courses of the total mesh stiffness are drawn up on one base pitch. The specific range of variation which is characteristic for the intensity of the pararnetricexcitation could be reduced by the three-dimensional flank correction from 3.3% to 0'.9%. The simulation of the dynamic behavior of the gear at various rotational speeds shows that the max- imum dynamic load factor Kv could be decreased from 1.17 to 1.06., This is achieved by depressing the maximum resonance peaks as well as the first and CIRCLE A-35 ON READER REPLYCARD second order Fourier coefficients of the 20, Gear Technology You need Nonnac's Fonnaster CNC Grinding Wheel Promer The Fonnaster gives you inspection guageacQlracy • Eliminates need for skilled operators as th unit is Normae guarantees an accuracy of! .0001" (.002 mm) automatic. Unit can be retro·fit for use with automatic infeed. from programed dimensions and provides less than .000 l" Now Nonnac offers you software Jar the Fonnaster (.002 mm) positioning error throughout total slide travel. fOIi automatic pliogramgeneration. Easily adaptable to meet your need. Normac has develop d optional software 'that runs on a The Formaster is designed to mount easily on nearly any type of personal computer in a question and answer format It aHows grinding machine. It is compact and lightweight and just a the user to enter part print specifications (including root and mounting bracket and only minor machine modiflcations are profile modifications) in standard gear nomenclature. The required for installation on most grinders. Inslallation usually program outputs :N.C code iloaded into the Formaster control takes one day or less. for wheel profiling, and also detailed printouts suitable for The unique features of Normae's gear profile analysis. This software contains logic for all Fonnaster .. types of root zone grinding, true involute and i exponential Rank modifications as weJlas "Proudn ss" .' The unit is totally seal'ed and air- purged for of the entire profile as indicated! in the K·diagram above. trouble-free operation and protection from coolant oontamination. Let us show you what the Fonnaster can • Double nut pre-loaded roller screws do for your grinding operaUons transmit servo motor to slide movement. If you would like. we can arrange a demonstration • Optional rotary diamond dressing wheel of the Formaster for you. CaU or write us today. attachment provides greater accuracy and e'liminates error caused by single point diamond wear. Fonnaster users have reported lncreased produdiionand greater aoourag-. • Increase in wheel: li~eof up to 8·10 times NORMAC, INC. 'longer than with other units. P.O. Box 69 '. Production increases up to 20% in the first Arden, N.C. ,28704 two weeks of use. (704) ,684·1002 • Set-uptime is reduced to 80% of what it P.O. Box 207 used to be with improved surface integrity Nol1hvllle, MI 48167 of the parts. (31,3) 349·2644 CIRCLE A-19 ON READER REPLYCAJ~,I:) mesh stiffness course. The corrected gear shows several op- ensure that a minimum contact ratio of Clear improvements of the dynamic timized characteristics. These are 1,.10 is achieved; thereby guaranteeing a: and bearing behavior can be achieved also • a much smoother mesh stiffness curve smooth operation, free from kinematic by geometrically simple correction forms. with a smaller specific variational and dynamic disturbances. (See Fig. 7.) Here the results of the tooth range, contact analysis ofa loaded helical gear • a reduction of the force levels at the Conclusions for the corrected and the uncorrected case beginning and end of engagement, A method is given for optimizing are illustrated. The tooth flank correction • a lower load reception gradient, various forms of tooth flank corrections consists of longitudinal end reliefs and two .' a.well-balanced and lower pressure with regard to the load carrying capacity involute profile corrections, The relief course. and the running behavior of spur or lengths and values were optimized in the The unmodified area. on the middle helical gears. Two examples for corrected method described previously. flanks of the corrected gear is sufficient to gears 'confirm that it ispossible to optimize flank corrections at the designing stage and to make empirical attempts un- necessary, Even when flanks are corrected by sim- ple geometric modifications which do not require advanced manufacturing meth- ods, the gear characteristics can be im- proved so that an increased transferable power and a reduction in noise emission is achieved. A more complex correction topography allows for an optimum con- formation with the gear geometry and the engagement conditions, while, at the same time, keeping the correction topography within manufacturing restrictions. It becomes dear that this new calcula- tion makes it possible to influenceall essential effects which determine the dynamic and load carrying behavior in a specific way. References: IG.···, 1. SALlE. H. 'Optimizing the Running Behavior I _ I of Cylindrical Involute Heavy Duty Gears." Dissertation, RVVTH Aachen, 1987. (Printed in Cerman.l Eliminate Carburizing P!roblems, 2. TESCH, F, "The Defective Tooth Engagement and its Effect on. Noise Emission," Dissertation. RWfH Aachen, 1965. (Printed in.German.) Induction AdY,antages Commercial Processing World Wide L,eaders 3. MOLLERS, W. "Parametric Excitation of Vibrations in Cylindrical Gear Pairs." Disser- '. Greatly reduces distortion • COmplete high tech lab Inductoheat IS the leader tation. RVVTH Aachen. 1982. (Printed in '. Minimizes part finishing require' IrI inducuon hardening . • ln-house, high powered radio German.) ments frequency technology. • Facilitates cellular and in-line • Full production quantmes- large 4. MAAG 'Topological Modification - A fur- processing Now we have developed ther Step in. Gear Technology." Technical Paper or small advanced the process of contour • instant start-u p and sh ut down ES-42.2. Maag-Zahnrader AG, Zurich, 1985. '. Individual part quality control • Development or heat treat gear hardening. The experience (Printed in Gennan.) • Exclusive process Signatu re patterns and specifications and resources of our international monitoring - • Prototylles for test purposes opera~ons have been combined 5. WECK, M., MAUER, G., SALrE, H. "J-D· to :brmg you this new technology. Tooth flank Corrections - Improving the Calltooay and we will show you the Bearing and Running Behavior of Gears." advantages of contour gear - hardening and how it Jlldustrie-Anzeiger, Leinfeldell·Echl'r!rdingllll, will benefit Vol. 109, No. n. 1987, pp. 3()'3J.. (Printed in • you. Gennan.) I _ 6, NEUPERT, B. "Calculation of Tooth Forces. NDUCTOHIIAT Pressures and Stresses of Cylindrical and Bevel 32251 North AVIS Drive Gears." Dissertation, RWIH Aachen. 1983. Madison Heights, M14!W71 (Printed in Cerman.) (313) 585-9393 TWX 61 (1.232·5208 1'~4.6<'91 FAX(313)SS9-1062 7. WECK. M.& REUTER, W. "Simulation of Tooth Plank Grinding Processes for Finishing Manuf.c1urlng PI.ntl In: Spur and Helical Gears." Paper for th 28th .. Australia. Belg!um • Brazil., England • France ,. India 'II' Japan. Taiwan Conference - Gear and Gear Cutting Machine • U.S.A. 11.11,TX) .' West Germany Research, WZL RWTH, Aachen, 1987, (Printed in Cerman.) CIRCLE A-37 ONI READER REPLY CARD 22 Gear Technology Enhanced- Product Performance- Through CBN Grinding ,Glenn Johnson and EmestRatterman,GE Superaibrasives, Wo.rthington, OH Abstract: The' extraordinary physical and thermal SIIlUAl SUI¥Jt.CE sm£SS ,IN properties of CBN abrasives area primary fac- AI$I_ r~g :~U%~~~N~I::8 tor in !:hegeneration ofbenefidal residual com- i5 ... I I --- , _. pressi.ve stresses, These residual stresses have \ \ --- ~ I I . - - - ,iI favorable influenceOI\, the fatigue life of com- ... 11 .1 lLj--W , \'tJiMooI--.- .... --La .... - ponents ground wi th CBN abrasives, This art- .. \ ~-...... icle suggests 'thaI, the effects of CBN grin.ding ..r~ .", I 1 .... "" II +_. \ should be factor d into the original design of 1 \ '" I'''' _ COtOTlOOIt '- highly stressed, fatigue-prone drive train '1 \ ~GJn:I ,1 ~cooova..- \\ ccmponents, - ~IIIIIIIQ.W--' , ~..l- I f-- .- '\\ .-- -DIh -.. -, -- I""- Inlrodurtion f-- '" ...'" '" QPTU --.... r.. '" " ... Modem manufacturing processes have ..''''' f " -" ~ ... ~~- become an aUy of the product designer in - .. I;u --- .. -.. I pmducing higher quality, higher perferm- .- , I .;" , -- l1li ...... ins components in the transportation in- • "" .. dustry. This is particularly true in grinding; ... "" ,.. ... systems where the physical properties of CBN abrasives have been applied to irn- Fig. I. - Residual stresses in SAE4340 after grinding .Fig:..lb - Typical residual' tresses In SAE September/October 1988, 23 I ·, - BOO1'H NO. 6691 I .~~ I ~IM·'TS·.·- - The world of manufacblring technology I• \ It takes technological virtuosity to put Why CIMA's Flexible Systems giving our customers more choices together a flexible manufacturing sys- Can Give Virtuoso Performances. in such things as controlJers.and tern like:the one you see here. And At the heart of our systems is our gear loading systems. while we at CIMA USA are perfectly hobbing equipment: That's where Our most virtuosic feat of all, willing to take a bow for its t1awless virtuosity begins. Evety machine though, happens when YO!J need a perfol'mance, we think a lot of the cred- CIMA USA 'builds is geared for the complete customized systtfm. You it belongs to our great Italian heritage. maximum speed, accuracy and effi- can rely entirely on our engineering And to men like Leonardo da ~nci, cieney that has earned our company wizards to design, engineer and install who knew how to put together a design _bravos from around tt}e world. But a turnkey system around any CIMA on a grand scale wrth little more than we've contributed much more to the machine. A system complete with a compass and a few drawing tools. art and science of cutting gears. By gear shavers, grinders, deburring, and The original gellr viriuoso, l£onartW an Vinci rnA:dt inge~ niDus com binI/lions of gears, cranks and pulleys: work in con- cert, to perform the same USt/ul tasks we put them 10 today, , computerized inspection machines. labor costs and enjoy the heights of master the art of increasing produc- CIMA alone has the flexible netmrk productivity like never before. tivity without increasing labor costs. to put ij all together for you. And! the All you really have to do to have And make an old master's dream virtuosity to make ~ IMJrk, virtually qear virtuosity on your manufaduring come true for your company. arr by itse~ floor is lift your finger. Anj push a~few The' ,Real Virtue ot Our Systems buttons. Our number is (804) 794-9764. C M USA is Higher IProductivHy. Call us today and t,ell us exactly wnat Purring Powerfulldeas In Gear. 'I.J\ia think even lronardo would applaud your needs are. Or get more infor- Division of G.D. P.M., Inc. I the fact that ocr systems IJIJOrkfaster mation by circling our name on the 501 Southlake Boulevard il Richmond, Virginia 23236 and better without a lot of human reader card. (804) 794-9764 . supervision. So you can save on CIMA virtuosity. It can help you Telefax (804) 794-6167· Telex 648-4252 CIRCLE A·IO ON READERREPLY CARD ing" will generally leave a residual tensile cedures, such as low stress grinding (LSG), in Table I. The density (e), thermal con- stress in the surface ofa workpiece, such nital etching or shot peening, may have to ductivity (k), and specific heat (c) are as AJSI 4340, unless special precautions be employed in certain cases to overcome listed along with the calculated value of are taken to avoid this -phenomenon. (6) It these inherent problems. thermal djffugjvity ..Thermal diffusivilty (a is also generally agreed that these residual That the crucial importance of the thermophysical property, k/,,,c), is the tensile stresses are created because the specific thermal properties of aluminum ratio of heat conducted versus the heat ab- "grinding" process rapidly increases the oxide grain have been overlooked is sorbed in a body. (7) In transient situa- temperature of the ground surface, which understandable; but in view of the grow- tions, such as the grinding process, N process, any single abrasive grain in Model ,6150 - 60" dla.., 12N or 20 face will be Model G250 - 98~ dia.• 21'" or 30'" face contact with the work for onJy80 micro- Im,proved Cutter . seconds. The analysis examinesthe tem- Box permits exclusive U.S. agent resharpening perature distribution in both an abrasive I without h Temperatu,e "C II Aluminum OXtde Their work suggests that 6,3% of the heat generated in aluminum oxid grinding goes Fig. 2a - Finiteelem.entanalysis model of heat transfer Fig. 2h - Comparison of temperature distribution down into the work, while in CBN grind- between workpiece and abrasive. after BOJ.LSeCof contact between workpiet.:e and abrasive for three types of abrasives. ing, only 4% goes into the work. They have concluded that chip fermadentakes ,...... , -- . place ata much lower temperature in the 121.7 12 Diamond C8N Experimental Investigation. Most investigators have conducted. em- Fig.,2b Fig. 211 pirical residual stress studies using some form of plain surface grinding and flat TtI:i!lle'll SELECTED PHYSICAL PROPERlIES OF CBN AND AWMINA ABRASIVES workpiece specimens. This in.vestigation will utilize cylindncally shaped specimens. In addition, most studies concentrate on the residual stresses obtained on only two' or, at the most. thr-ee sample surfaces fora given combination of abrasive grain. and grinding conditions. This study will. utilize a total of 93 individual workpiece samples, all ground with the same CBN wheel spec- ification ..This investigation is comprised of 'TRADEM:ARK OF GENERAL ELECTRIC CO the following steps: Ta'ble III 1. Sample preparation - Two sets of IDESC'RI:PTIONI OF GRINDING WHEEL ANDCONDIII,IONS steel cylinders, 1" in.diameter by SII long, USE.IO'IN SAMPLE PREPARATIION have been carburized, heat treated and quenched to produce a hardness of Rc 62-64. A total of 40 cydinders of SAE 8620 and 501cylinders of SAE 4620 have been ac- cordingly prepared. Each sample cylinder has been cylindrically plunge ground with a one inch wide wheel containing CBN abrasive. The details of this procedure are shown in Table II. Fig.. 3 illustrates the dimensions and configuration of the fin- ished samples. 2. X-ray diffraction residual stress analysis - The surfaces of the samples pre- "TRADEMARK OF GENERAL ELECTRIC COMII'.'~Y. USA pared in. Step 1 above have been analyzed September/OOtober 1988 27 using established techniques of x-ray dif- ,Ii fraction analysis. This analysis has been I conducted on both the unground and the ground surfaces of each sample. Peripheral and longitudinal stresses have been selec- tively measured on the unground surfaces .. The peripheral stresses in the direction of t=ql...... -- I!----f"--I--r---IL---='EJ grinding and longitudinal stresses at right e.m GOO"ND 1007 01" I angles to the grinding direction have been n,", measured on the CBN ground surfaces. 3. Effect of grinding depth on residual stresses - The amount of material to be removed in production grinding opera- fig. 3-Details of SAE 4620 and 8620 cylinders used to compare residual stresses - as-heat treated and tions cannot always be precisely con- CBNground. trolled. The slight distortions incompo- nents which have been heat treated can lead to variations in grinding depths in fI TYP TVP ~!=---"-I such cases; thus, detemrining how residual I stress will vary as material is removed from 0 the unground, heat treated surface is im- -1 -1 w~ -I ~~ !a portant. Therefore, another set of cylin- 6 01 0 0 0 n. Q - a ~ ;:! rr ~ ;:: drical samples have been plunge ground to ~ oJ a range of finished diameters in. order to determine this eff·ect. The configuration of 1 1 1 1 :-'- such samples is illustrated in fig ..4. The ..., grinding conditions used to produce these surfaces are !he same as shown in Table II except for the use of a 0..5" wide wheel. fig. 4.- Details of SAE 4620 and 8620 cylinders used to determine effect of CBN grind depth on residual stresses. 'TRUE ,DIMENSION GEAR lNSP'ECTION Provides actual over hall/pin measure- ment of any helical or spur gear or spline with- out the need of costly setting masters. Provides vital S.P.C. information. Gage Divi'sion CAPACITY: United Tool Sup'ply 9" O.D. 8" I.D. 851 OHIO PIKE. CINCI'NNArTI, OHIO 45245 • (513) 752·6000 CIRCLE A-33 ON READER REPLYCARD 28 Gear Technol'ogy CIRCLE A-2 ON READER REPlY CARD .Fig. S - Residual stresses in SAE 4620 bef,ore and after grinding with CBN abrasives(S1 samples) . •..~------+------+------+------11 Fig. '] - Residual stresses in SAE 8620 after CBN grinding to various depths below hardened surface . .. PEJlll'HEML I • I.OItQ!TUOIIU!. ~ • I -'1.'5 ·30 J r -4.5 ~ r-, I ~ -10 I ~ -~ ---... iil ~ I w -110 -- '" -, j -"" -1m I I -1:15 o ------2. - Fig. 6 - Residual stresses in SAE 8620 before and after grinding with CBN abrasives (40 samples). -.. • _L L r 1 • LDNOITUlllNAL I I -30 I ~ -tS ~ I m -'" 1 '"to -",'5 -, T ~ -IfO iil -, w I I '" -"" I .. liQ -135 -- J ""!" _t5O • III: 10 !2 14 .. DEPTH BHOW OR'GINAl SURF'''CE~~N)I 1()1 Fig. 8- Residual stresses in SAE 4620 after CBNgrinding 10 various depths from as-hardened surface. 30 Gear Technology Results Tne results of this work are detailed as follows: 1. Residual stress analysis - The of). ==P unground and ground SAE 4620 resul.ts in - ±Z sigma limit bars illustrated in Fig. 5. r- Analysis of these results using the student , -75 11 'T' test for signiticanee reveals that CBN grinding has had the following effects on -100 'I the residual stress levels of the ground fl samples: -125 4620 - 95% confidence that the -150 longitudinal stress is increased from - 96250 PSI to -148600 +/ - 5900 PSI, -175 and the peripheral stress from- 61800 PSI to - 88UO+ I -5200 PSI. I 8620 - 95% confidence that the -225 i longitudinal stress is increased from 6 DEPTH (IN • '0..31 - 45200 PSI to -133300 +/ - 3600 PSI, and the peripheral stress from -43300 PSI to - 92700 +/ - 3900 PSI. Fig. 9~ - Mid-flank radial residual stress analysis. . 2...Effect of grinding depth on residual stress - The effect on both the SAE8620 and SAE 4620 ground samples are shown graphically in Figs. 7 and 8. Inthe case of both workpiece types, the consequences of 0 CBN grinding dearly have been to further increase the residual compressive stress as -25 r- thetotal grinding depth is increased. -50 (( Discussion -15 i r::" -----' The general character of the residual ~ Ul -100 stresses which result from CBN grinding in W ill 't ... I OJ 1/ , this work is in agreement with previously a: , t; -125 reported work. This investigation, how- -' ~ ever, has shown dearly that the residual c -150 iii w compressive stresses created by CBN a: I! -175 grinding are additive to the inherent i residual. compressive stresses found on as- -200 , heat treated surfaces (Figs. 5,6,7,8). This I isa completely new finding which should -225 ! '8 10 trigger further investigation. OEPTH (IN " '0-31 Recently we obtained samples of auto- motive transmission gears ground with an electroplated CBN wheel. Five gears were Fig. 9b - Mid-flank radial residual stress analysis. selected at random during the course of grinding a. run of approximately 250,000 gears. The midflank residual stress in the AUTHORS • radial direction has been analyzed, and the GLENN JOHNSON is superoisor at GE plication Techno.logy Communications at GE results of these analyses are shown in Superabras.ive5 Pacific AppUcation Develop- Superabrasive5, Worthington, on. He has Figs, 9a and 9b. The residual stress distrib- ment Center in Gotenba, Japan. He is respon- worked in the helicopter and nuclear industries ution is developed to a depth of.'OOB" sible forprograms associated with the applica- and has over twenty years'experience in the ap- tion of CBN and diamond in grinding of metallic below the flank surface. The results show plication development of superabmsives. He has and nonmetallic materials. Prior to joining GE been manager of GE Superabrasives Applica- a remarkable consistency of compressive Superabrasive5, he worked in process develop- tion Technology Operations and is now respOI1- residual stress level. at the flank surface. ment at GE Mining Products Division. Mr. sibl'efor .applying new and conventional 001'11- in The small subsurface profile variations lohnson has a degree in mechan iealengineering munications technologies to superabrasive the residual stress distributions will be due from Penn State University. educational needs. Mr. Ra.ttemu111 holds Q to variations in the heat treated profiles. degree in aeronautical engineering from the Overall, these results again confirm the in- ERNESTRAITrnMAN is Manager - Ap- University of Cincinnati. September/October 1988 31 Mitsubishi Shapes the World of Gears ... What can Mitsubishi do for you? Mitsubishi can support single source hardware supply, engineering support you in all fields of gear manufacturing technology. and responsibility. Look for Mitsubishi builds gear hobbers, gear shapers and gear Mitsubishi, the world leader of gear shavers. All CNC controlled. Mitsubishi not only builds manufacturing technology. gear machinery but also manufactures various kinds of For more information, call cutting tools. TiN coated gear hobs, shaping cutters and (312) 860-4220, NOW! shaving cutters, etc. With Mitsubishi, you can get a • ~'I Come and see us at McCormick Place North ,,- I'" '. fo.rmore exciting details and demonstrations. ;:;~ I··MTS- - (0)(0) ~~.~VJ' I -_. ~ See UI It :Elooth No. 63112 McOormlck North Mitsubishi Heavy Industries America,. Inc. "MIITSUBIS!H,I 873 Supreme Drive, Bensenville, IL 60160 Phone: (312) 860-4220 ~- HEAVY INDUSTRIIES. LTD. 5-1, Marunouchi 2-chome, Chiyoda-ku, Tokyo, Japan MitsubishJ Intellnational Corporation Ca.ble, Address; HISHIJU TOKYO 873 Supreme Drive, Bensenville, IL 60160 Phone: (312) 860-4222 CIRCLE A-a ON READER REPLYCARD Flg.10 Honda Motor Co. has reported that the use of CBN grinding has made it possible 30 to design and manufacture final drive gears of reduced weight and which operate at III lower noise levels. These attributes derive z 25 0 from both the increased uniformity of 3 ,.i tooth form and beneficial residual stresses. tt: :3 20 ~ e Conclusions III This investigation has established that: .. 15 d > • CBN grinding of carburizedand hard- 0 Z ened parts can impart additional residual ~ 0: , 0 compressive stresses to the part surface. These stresses may be from as littl as 30 % greater to as much as 250'% greater 05 than the heat treated surface stresses. • CBN grinding develops an increase in compressive residual stresses as grinding TEST , 2 3 -4 2 3 • I 2 3 progresses from an as-heat treated sur- • CONVENTIONALL '( "CBNGROUND CBNGROUND PROCESSED FLANKS ON LV FLANKS & FILLETS face into the case. The generation of TEST GEAR DESIGN SPEC1FI(;ATIONS compressive stresses has been shown in HYPOIO DESlGN TESTS WERE CONDUCTED WITH TOROUE APPLIED WAS 70% Figs. 7 and 8 to be independent of the 4288DP GEIoR PAIRS 'IliST ALLEO IN OF FULL AXLE TORQUE RATING 11.4SRATIO "-IlLES USII'IG A 4-SOUARE 'CONYENOOI References ing of Thin Workpieces With Superabra- pressive Residual Stress Resulting From 1. NAVARRO, N.P .. "The Technical and sive 'lNheels", ClRP Annals. Vol. 36, CBN Grinding", SME Second lntema- Economic Aspects of Grinding Steel With 111987. tional Grinding Conference, 1986. BORAZON Type II and Diamond", 6. HELD, M., J.F. KAmES. 'The Influence 10. DODD, H.P., K.V. KUMAR, Society of Manufacturing Engineers, of Untempered And Overtempered 'Technological Fundamentals of CBN MR70-198, April, 1970. Martensite Produced During Grinding on Bevel Gear Finish Grinding", Society of 2. CHUKWUDEBE, t.o. "Grinding The Surface Integrity of AISI 4340, HRC Manufacturing Engineers, Superabrasrves Superalloys With 130RAZON GIN", Steel", Milton C. Shaw Grinding Sym- '85, Chicago, IL, April, 1985. Proceedings DWMI, November, 1975, posium, American Society of Mechanical 11. KIMMET, G,J., "CBNFinish Grincii!\gof Chicago, IL. Engineers, Miami Beach, FL. 1985. Hardened Spiral Bevel and Hypoici 3. TONSHOFF, H.K., "Influence of the 7. lNCROPERA, F.P., D.P. DEWITT, Fun- Gears", American Gear Manwactur rs Abrasive on Fatigue in Precision Grind- damentals of Heat Transfer, John Wiley Association. 1984. ing", Milton C. Shaw Grinding Sym- & Sons, 1981. 12. YOKOGAWA, K., "Grinding Perfor- posium, American Society of Mechanical 8. RAMANATH, S., M.e. SHAW, "Role of mance of CBN 'lNheel - Report #22", Engineers, Miami Beach, Fl, 1985. Abrasive and Workpiece Properties Japanese Society of Precision Engineers. 4. ALTIiAUS, P.G., "Residual Stresses in In- Relative to Surface Temperature and Spring Conference, 1987. temal Grinding", Industrial Diamond Residual Stress", Proceedings 14th North 13. PRIVATE COMMUNICATION - Hon- Review, 3/85. American Manufacturing Research Con- da Motor Co. to General Electric Com- 5., MATSUO, T., H. SHlBAHARA, Y. ference, May, 1986. pany, Worthington, Ohio, USA, July 2, OHBUCHl, "Curvature in Suriace Grind- 9. JOHNSON, G.A., "Beneficial Com- 1987. September/OCtober 1988 33 - - BACK TO BASICS ... - -- -- Involute Curve Fundamentals Involutometry Over the years many different curves have been considered for the profile of a gear tooth. Today nearly every gear tooth uses an involute profile ..The involute curve may be described Harlan W. Van Gerpen, P.E. as the curve generated by the end of a string that is unwrapped C. Kent Reece, P.E. from a cylinder. (See Fig.1.) The circumference of the cylinder Van Gerpen-Reece Engineering, Cedar Falls, IA is called the base circle. Following are specific features of the involute curve. • A perpendicular to the involute surface is always tangent to the base circle. • The involute surface is a uniform rise cam (equal rise per in- crement of rotation). • The radius of curvature of an involute surface is equal to the length of the tangent to the base circle. • The same force applied perpendicular to the involute surface of a gear tooth at different tooth radii results in the same tor- que on the gear. Because the involute surface is a uniform rise cam, any in- volute surface imparts unifonn angular motion when operating against any other involute surface. To illustrate further, con- sider two pulleys with a crossed-belt drive as shown in Fig. 2. If we place knots in the string, then the knots will generate in- volute curves as they leave one circle and approach the other circle as shown in Fig..3. If the knots are spaced close enough, there will always be knots on an involute surface when they are not on the circle. These generated curves can then be considered as teeth on the gear. I'ig.l-lnvolutecurve. .. Fig. 2-Crossed belt drive. Fig. 3 - Multiple involute curves. 34 Gear Technology Of course, there are otherconsiderations. The distance be- angle the line of action makes with a perpendicular to the center tween knots must add up to an integer around the base circle, line of the gears is called the pressure angle. The pressure angle and the knots must be far enough apart to provide a structure is defined as B di ase ra IUS for the tooth. When these criteria are met, the distance between Pressure angle 0: = Acos (1) knots is called the base pitch ..Any two spur gears that have the Pitch radius same base pitch can be meshed together. (See fig. 4.) Just as the pitch radius with another gear is unique, so the Just as the diameters of the two, pulleys in the crossed-belt pressure angle with another gear will be unique. drive establish the ratio of the two shafts, likewise, the This may be confusing until one is reminded of the crossed- diameters of the base circles establish the ratio .of the two gears. belt drive ..The cylinders would rotate in the same manner, and The contact between two involute surfaces of two gears the knots in the string would generate the same involutes in- always occurs on a line that is tangent to the two base circles. dependent of the center distance between the two cylinders. The This tangent line is called the "line of action" ..(See Fig. 5.) This crossover point of the line of action across the center line will phenomenon results from the from tact that aperpendicular to result in different pitch circles, and the pressure angle will the involute surface is tangent to the base circle. change as the center distance is changed. The force along the The line of action will cross the center line between the two line of action will remain the same; it is the torque divided by gears. This intersection is called the pitch point. If we draw the base circle radius. As we will see later, there are considera- circles 'on each gear using the gear centers and passing through tions of tooth thickness and contact ratio which restrict the the pitch point, we have pitch circles. These pitch circles may be considered as cylinders which would roll with each other by friction with no slippage. The distance between the center lines of gear teeth on either pitch circle is called the circular pitch. If we divide the circumference of the pitch circle by the number of teeth in the gear, we get the circular pitch. These circles are important in the calculation 'of tooth thicknesses because it is here that the two tooth thicknesses, plus the backlash, must add up to the circular pitch. Each gear has two distinct pitch drdes to be considered. The gear win have one pitch drdewith its cutter and another pitch circle with the mating gear. If it is the middle gear in a string .of three gears,it may have a different pitch circle with each mating gear. The pitch circles of mating gears will change as the center Fig. 4 - Base pitch of a.gear and a rack. distance is changed. ITthe center distance is increased, the pitch circles will get larger, and the circular pitch wilJ increase with the same tooth thicknesses as before. The backlash will also in- crease. A gear does not have an operating pitch diameter until its mating gear and center distance are specified. Since gear teeth must make contact along the line of action, and since the contacting Iorcesare parallel to this line, there is a force which acts to push the tWD gears' centers apart. The AlITHORS: HARlAN VAN GmPEN has 28 years of experience at tile 10hn Deere Product Engineering Center i'l Waterloo, Iowa, where Ile served as manager of tedmica[ services tmd a pn'ncipaJ engineering specialist. He is a licensed professional engineer in the State of Iowa and a member of the ASAE where he is a mamber of the Research Committee and the In- strumentatioPl and Controls Committee. He is also a member of the SAE Education and Human Factors Committees. Mr. Van Gerpetl holds tI masters degree in electn'cal engineering from the University of Illinois. CARROLL K. REECE worked for the John Deere Product Engineer- ing Center, Waterloo. Iowa, as an engineering design analyst and super- visor in the mechanical elements department. He has seroedior 25 years on the AN51-B92 National Spline Standards Committee and for 18years 011 both the.lSO TC 32 International Spline Standards Committee and the ANSi-B6Nationa} Gear Standards Committee. He holds a masters degree in agricultural engineering from Kansas State University and is a licensed professional en.gineer in the State of Iowa. Fig. 5 - Line of action and pressure angle. September/October 1988 3S range that the center distance may be changed. In very large power applications common practice dictates The insensitivity of the involute action to the changing of the the use of two helical gears, side by side, with opposite angles center distance of the gears is a distinct advantage over other of the helix (hand) so that the end thrust is neutralized. Usually tooth profiles eonsidered in the past As shafts deflect due to there is a small space between the gears to facilitate manufac- changes in. load, the involute action is unchanged, assuming ture and also to give the tooth flexibility for load sharing. sufficient contact ratio still exists. Helical gears have involute tooth profiles similar to those in The involute profile is the path of the end ofa string un- spur gears. Each gear will have a base cirde and a line of action wound from a base circle. The design of the gear involves like a spur gear ..[f we observe the gear from an axial posit ion, establishing the ends of this curve. At the beginning of the curve the gear contacts will occur along the line of action. However, there must be some sort of fillet blend to the root and the in- an entire tooth will not make contact with the mating gear in- volute of the adjacent tooth. It is not good practice to plan on stantaneously, as does a spur gear tooth. The contact may start using the involute near the base circle because its radius of cur- at the tip of one end of the tooth and proceed along the length vature is very small and changing rapidl y. It is very difficult to of the tooth, with the first point of contact moving down the manufacture the tooth accurately near the base circle. tooth before the other end of the tooth makes 'contact. The in- However, this end of the involute profile must be defined ac- stantaneous load contact will be a diagonal line across the tooth curately to avoid problems of interference with the tip of the face with a different radius as we follow the contact across the mating gear tooth. The "form diameter" is used to specify the tooth. For very wide gears the pattern may be repeated several mot end of the involute profile. times across the gear as several teeth, with their helix angles, The tooth tip end of the involute profile can either extend to contact on the line of action. the outside diameter of the gear, or if a chamfer is provided, it With helical gears we have a second contact ratio to consider. ends with the start. of the chamfer. A chamfer is used by many The contact ratio expressed by the length of the zone of action manufacturers to prevent damage to the involute surface from divided by the base pitch is still used, The second, called face nicks due to handling. contact ratio, is a function of the helix angle and the width of the gear. For smooth action both contact ratios should exceed Helical Gears one. More and more new machines use helical gears. The prin- The teeth on a.helical gear have a slight "twist" in the radial cipal reason for this is the demand for quieter gears. Not only direction ..lf a "pin" is placed in the tooth space, as is done when do we have an increased emphasis upon quiet machines, but measuring the distance across pins to determine tooth also most machines operate at higher speeds and higher loads, thickness, the pin will sit on a high spot and "rock". A ball must both of which increase the noise made by spur gears. A pitch be used for making this measurement with helical gears, which line velocity of 2000 feet per minute is considered by many to have an odd number of teeth. be the top limit Ior using spur gears innon-aircraft applications. To study the tooth geometry we have to consider two planes, The writers have observed spur gears operating quietly at pitch the transverse plane and the normal plane. The gear teeth line velocities of 600() feet per minute, but these were probably operate in the transverse plane, but the teeth are generated in very accurate gears. With the normal indust:rial production the normal plane. The tooth thickness and generating pressure tolerances, we can expect spur gears to be noisy above pitch line angle usually refer to the norma] plane. The base circles, out- velocities, of 2000 feet per minute. If this speed must be exceeded side radius, root diameter and contact ratio are determined in and quietness is preferred, then the design should incorporate the transverse plane. helical gears .. To determine the tooth thickness at a second radius requires We will introduce several new terms to describe physical the conversion of the given tooth thickness in.the normal plane characteristics of helical gears ..The first of these is the helix to the transverse plane. The tooth thickness at the desired radius angle. The helix angle is the inclination of the tooth in a is calculated in the transverse plane. It can then be converted lengthwise direction ..The helix angle results in the gear having to the normal plane again, taking into account the change of end thrust or an axial force during operation.and the support helix angle at the desired radius. bearings must be able to resist this force. There is no rule for the Helical gears may be cut with the same hob that is used. to cut best helix angle. It may be up to 45°. spur gears. The hobbing machine has the ability to position the The helix angle of the tooth depends upon the radius under hob at an angle corresponding to the helix angle. consideration ..Nonnally we quote a helix angle associated with For gears that must be cut with a shaper because there is no the generating helix angle. This relates to the pressure angle oJ clearance for hob runout, the shaper machine must be equipped the cutter and the generating pitch radius. It is more important with a "guide". The guide provides rotation of the cutter as it to work with the helix angle at the basecircle, For two gears to generates the teeth. Each guide has a specific lead, and the helix mesh properly and operate on parallel axes, they must have the angle generated depends upon the diameter of the shaper cut- same normal base pitch and the same base helix angle. ter. A large diameter cutter will generate a larger helix angle To determine the different helix angles on a.gear requires a than a small diameter. The shaper cutter must have nearly the familiarity with the term "lead". The lead of a gear may be de- same lead; "nearly" because the shaper cutter may stiill be usable fined as the width of a gear required for a tooth to wind one even though its helix angle is up to one degree different than the revolution around the gear. A large helix angle will result in a helix angle being generated. The cutter has enough side relief small lead. A spur gear may be considered as having an "in- to permit this. This should only be done for small lots .. finite" lead. When selecting shapers for a helical gear, the availability of 36 Gear Technology a guide is mast important because the cost 'Of a guide is many times that of a shaper, The mathematieal formulae for helical gears aile nearly the same as for spur gears. except for the indusian of a function of the helix angle In the equation, The formulae are basically set up to. wark in.the transverse plane (plane of rotation); therefore. u the tooth thickness and pressure angle a:rein the normal plane. they must be converted to the transverse plane. Gear Design Mathematics The tooth geometry 'Of a spur or helical gear may be de- scribed by seven pieces 'Of data. These are number 'Ofteeth, base diameter. 'Outside diameter. form diameter. root diameter, tooth thickness at base diameter and lead (and hand). The design of a gear involves the use of mathematical equa- tions to 'Obtain the above seven numbers. The mathematics in- valves extensive use of trigonometry . The designer must be Eq.2 familiar with the use 'Of angles in radians and the involute function. See us atl!MTS·88 As mentioned previously.the involute curve is the path of Booth #6691 the end 'Ofa string unwound from a cylinder. The fan owing relationships hold. * GEARS * DEBUHR & CHA F R quickly, accurately, inexpensively Tan a = a + () ...with rugged machines designed () =Tan a -a = iavo (2) to perform year after year ...in Cells, Fixed Mount or Portable The involute function is the angular displacement of a point on the tooth from its pointaf intersection with the base circle. applications This relationship is useful for calculating the circular thickness 'Of a tooth. When the thickness at a given diameter is desired. there is a corresponding angle or "pressure angle" of the tooth at that radius. By determining the pressure angle at a point on a tooth and then taking the involute of this angle, the angular shift of the curve from where it left the base circle is available. Now by multiplying this angle by the new radius, we have the length 'Of arc of the shift of the tooth pl'afi1e. By subtracting two times this angle from the angle on the base cirde for the entire tooth, and multiplying this resultant angle by the desired ramus. we get the circular tooth thickness at the desired radius. (See Equation 3.) TR = 2R I' !b...- inv,al a -= Acos I~l(3) 2'~ Good practice suggests use of the tooth base thickness as a [undamerual dimension on the gea:r because finding the thickness at any other radius 'Onlyrequires finding 'Oneinvolute. If the known thickness is at some other diameter, then it is necessary to have the involute at this radius, and then use the difference in involutes to determine the ·diHerence in known tooth thickness and the desired tooth thickness. CIRCLE A·12 ON READER REPLYCARD September/October 1988 31 Eq.J Eq ..4 It Is customary to use the circular dimensions for describing transverse plane is tooth thickness. The difierence between the circular and chor- dal tooth thickness is very small, usually less than one-tenth of the backlash that will be provided. However, in some PD = Z (5) mathematical analyses, such as following the path of a cutter, DP'COS~ chordal dimensions must be used to get the accuracy required. where Gear designers who work with "standard" gears usually have the tooth thickness at the pitch circle and then have to use two Z =Number of teeth involutes to find the tooth thickness at another radius. DP = Diametral pitch Given the arc tooth thickness and pressure angle in the plane (normal) of rotation of a helical gear at a given radius, to determine the i3 = Helix angle tooth thickness at any other radius, use 'the following formulae. (generating) R1· Casal Converting the norma] tooth thickness to the transverse R2 plane requires the following fonnula: T = Tn (6) (4) I Cos~ where where Rt. = Given. radius 'r, = Normal tooth thickness at. = Pressure angle at Rl (3 -" Helix angle Tl = Arc tooeh thickness at Rl T, = Transverse tooth thickness R2= Radius of unknown. tooth thickness ct2; = Pressure angle at R2 Note that the helix angle must correspond to the diameter on T 2 = Tooth thickness at R2 the gear corresponding 10 the tooth thickness. The helix angle is different for different diameters of the tooth. The lead is con- The above formula assumes all the data is provided in the stant for a gear and can be determined from a helix ..angle given transverse plane (plane of rotation), Most helical gear data is at a given radius by the formula: available in the normal plane because the cutter is dimensioned in this manner. Such data as tooth thickness and pressure angle are usually in the normal plane. 2'1!"R L = .. - (7) The formula for the pitch diameter of a helical gear in the Tan (3 38 Gear TechnologV -I lBooth #8'016, Ag. 6-Diagram 1,0 illustrate helix angle variation. where SET-UPS R .,. Radius of given helix angle f) ~ Given helix. angle TAKE l = Lead of the gear SECONDS Oncethe lead is known, the foll.owing formula may be used * 1'.NTERNAl~EXTERNAL. to find the helix an.g.Jeat 'the radius corresponding to, the given SPUR & HELICAl GEARS ,tooth thickness .. TO 20 INCIH,ES D,IA_METER JAMES ENGiIIN!EERING 2' r' (8) f) = Atan L. RJ 11707 McBean Drive, EI Monle, CA 9'1732· (6181442·2898 ( CIRCLE .A-1I11ON! READER REPLY CARD' where R- Radius of given tooth thickness L - Lead of the gear We' Can Solve Your' f) - Helix angle for R Distortion 1,..------' Fig. 6 is helpful in remembering the relationship between the and Wear lead and helix angle at different radii of the gear tooth. The lead Problems •. is constant for a gear, but the ciecumlerence depends upon the radius being studied .. • State-of-the-art gear hardening equipment The study of the beam stress oJ the tooth and the form. including 4 NATCO : diameter is done in the normal plane. Doing these studies in the submerged process L...- __ normal plane r quires converting to a "virtual" gear. The nor- machines and 3 AJAX gear scanning mal plane of a helical gear is really an ellipse. The point of in- machines. terest is the largest radius of curvature of the ellipse ..This radius • Specialists in hard- of curvature is used Ito construct a virtual gear. ening helical and The virtual gear is a simulated spur gear. Mathematically, bevel gears and the virtual, gear is derived from the helical gear by the follow- contour Ihar,dening of gear teeth. - ing relationships: .•We can 1001to meet any production needs. -· ] .- h di (R) Pitch radius (helical gear) .•Write for free ca,pa- Vrrtua pttc .ra JUS ,,= ------'~- (9) bilities brochure. Cos 2 (Helix angle) Amerlean MI !treatlngl 'Company Virtual number .of teeth (Z ) = "{;eethin helical gear (Z) (10) 1043 Ee.sl'62nd Slrllet Clevelal'1d. Ohio 44103 v cos3 (Helix angle) (216)4:}14492 FAX: (.216) 431·1506 Virtuall base radius (R ,,) == b (11) Virtual p.itch radius' Cos (PAnol1lliill) Specialists In ,Mlnln9 and Off- The-Road ,Equipment 01_ ,""," 'T~CD_., To 0brain the ou tsid.ediameter and root dia..rnel:er,their dif- CIRCLE A.-9 ONI READER R.EALY CARD The given pressure angle will usualJy correspond to the . Z1(T1 + Tz)- 2' 'K' Rl + . mvaz = . mval pressure angle of the cutter. To convert this into the transverse 2 . R1 (Z1 + Z2) plane the following formula is available. C1= R1 + R2 (13) Tan an Tan at =-- (12) Cos {3g Cosaz where an = Normal pressure angle (38 = Generating helix angle Rl = Given radius of first gear at = Transverse pressure angle R2 = Given radius of second gear Z1 = Teeth in first gear Another calculation that is important in designing a set of Zz = Teeth in second gear gears is determining the center distance for two gears to be at = Pressure angle at R1 and r2 meshed 'together. Again, it isoruy necessary that the gears have a2 = Pressure angle at meshing position the same base pitch for the involute action to be proper; Tl == Ar,c tooth thickness at R) however, there may be interference with the tips of teeth going T2 = Arc tooth thickness at RZ below theform circle. The tips of the teeth may also bottom out C1 = Center distance for al in the root of the mating gear if the members ·of the pair were C2 - Center distance for 112 not designed together. If the gears are helical, they must have the same base helix angle. If they are external gears, the hand The previous equations are useful for determining the center of the helixes must be opposite, assuming they tum on parallel distance when run with a master gear. This is a tight mesh shafts. operating condition. If two gears are to be operated together, Given tooth proportions in the plane of rotation for two then the desired backlash may be added to the tooth thicknesses helical gears, to determine the center distance when they mesh in the equation, and the corresponding center distance derived. tightly, use the following formulae, Technical Education Semin,ars Your technology has to be the best. To help vou, AGMA is initiating a series of seminars - each coneentratlnq on a de- tali1ledaspect of gear technologlY. Regist,ration is limit,ed to preserve classroom interaction, so reserve your seat nowl Topics, L.ocaUons and Speaken for 1988 - 11989Include: Nov. 9 Controlling the Carburiziing Process - Cincinnati, OH - by Roy Kern, Kern Engineering Co. Dec. 7 Rational Loose, Gear Quality Requirements (using AGMA 2000 properly) for Specifyer and Purchaser - AGMA Headquarters - by Chuck Schultz, QuaKer Ci,ly Gear, Iinc. Jan. 11 Speci,fyingand Controlling the Quality of Shot Peening - Cincinnati, OH by Jim Daley, Me,tal'improvement Company March 7/8 Source Inspection of Loose Gears from the Customer's Standpoint - Rochester, NY by Robert E. Smith. R. E. Smith and Company May 2 Gear Math at the Shop Level for the Gear Shop Foreman - Olnclnnati. OH by Donald McVittie, Gear Engineers., Inc. June 6 Specifying and Verifying Material Quality per AGMA Material Grades - AGMA Headquarters by Rocky Andreini, iEarle Jorgensen Company For Further Inf'ormatlon C'o,ntact eill! Daniels, AGMA. Headquarten 1500 King Street, S,uUe 201. Alexa,ndir,ia, VA 22314 (70.3) 684-0211 CIRCLE .A-27 ON READER REPLYCARD 40 Gear Technology Ference from the pitch diameter is maintained in each plane. The contact ratio expressed by the length of the zone 0.£ action divided by the base pitch is sHUused. The second, called face Contact Ratio contact ratio, is a function of the helix. angle and the width of If two mating gears are to provide smooth rotation, it is the gear. For smooth action both contact ratios should exceed necessary that there be a continuing 'contact on the line ofac- one. The formula for face contact ratio is tion. A second pair of teeth must mesh before the first set separate. This relationship is described by the "contact ratio". - F'Tan~ CRIf ---- (15) The contact ratio is the ratio of the length of adiveengagement race p of the gears on the line of action to the base pitch ..For spur gears where the contact ratio must exceed one and may exceed two for gears where quiet operation is needed. F = Gear face width The following figure and equations show how contact ratio f3 = Helix angle at pitch point is determined. P ,= Circular pitch (transverse) C~ace = Face contact ratio 2 R2 CR =YF:fJ1-R bl + YR52- b2 - C'Sincx (14) p. Cos e Generation of Gear Teeth While we discussed the forming of the involute surface ofa where gear tooth as identical to the curve of a string being unwound from a base cylinder, the following is descriptive of how a gear Ral= Outside radius of gear 1 tooth is actually generated. There are two main types of cut- Ro2 = Outside radius of gear 2 ters used to generate gears, ho bs and shapers. First consider the C = Center distance "shaper", This cutter closely resembles a gear. One edge is sharp eR = Contact ratio and, by gradually meshing with a blank as it moves back and RbI = Base radius gear 1 forth axially , it will cut a mating gear. The blank should be the Rb2. = Base radius gear 2 appropriate diameter, and the two centers must be externally 0( = Pressure angle geared together with the right ratio. P = Circular pitch A second type of cutter is the "hob". Before discussing this cutter we must discuss the "basic rack". A basic rack is a seg- With helical gears we have a second contact ratio to consider .. (continued on page 45) •• r~IFROMAG ~ 1)-5.758 Frondenberg /,Ruhr The first fully programmable CNC keyseating machine .. Hliglh speed iinducti,oln ge'lar hardelninlg' cuts CIOS,ts, ,Bind elim,in,ates car'burizin,g Unlike currently availableR.F. systems, ULTRA-CASE iUILT'R:A-CASE:: 50 iKHz requires less than normal or no pre-heat cycle which often affects core hardness and somejimes can • IEUminates ca1rburizlng'. cause addit,ional distortion. • Is ,easily i.ntegr:ated ilnto your gear manufacturing ,ce'lll. Let Iindustrial Eh:lctric Heating put Ul:TIRA-CASE to • IEUminates ,dimensionall distortion Iprobl'ems. work for you. Send us a sample part or let us make a Iproduction run. We can even handle some of • Reduces overall manuf,acturin,g eosts. your heat treating jobs linour plant, For more • P,rocesses a wide variety of gears .. information, wflite to: • UtiUzes proven high performance technology. Industrial Electric ,Heating!,P.O. !Box 991, Warren. Ohio 44482: or call (216) 372-8506. The newlEH 50 KHz solidi state ULTRA-CASE telex 98 2482, tax (216) 372-8644. System offers you surface hardened gears at production line speeds. ULTRA-CASE enables you to harden 'gears twice as fast as any other comparable induction system with less 'equipment and at lower cost. Visit us at ASM Heat TreatIng Show in 11111=1-1 Chicago - Booth 1925. IlnduslriallEleclric Healing AN AJAX MAG,NETHERMIC G,ROUP' COMIPANIY CIRCLE A-3 ON READER REPLYCARD 11heCNC co:ntroilled HNIC-35 Wo'rm and 'Thread Grin,der Klingel'nberg introduces the fastest ... most accurate The HNC·35 is a.vailable with a mechanical dresser or complex worm and thread grinder ,available ... any- an optionalll'CNC dresser; for specialll forms and Uanlc:s. where in the worl'd'JThe CNC ,controlled HINC·35 stores Whether you Iproduce, small! quantities, or long IProduc- data for up to '960 comple.x sbapss, reducing set-up tion runs, the FAST set-up ... FAST cycling . ,.. HNC· ' ' to 110-15minutes ... ,a fraction 01 the time required with 35 win improve your worm and thread Iproductiivity. mechanicaUy contmlled grinders. For additional information and a copy of our 'catalogl, The HNC-35 is versatile too. It lperforms ,cree,p-feed contact: Klingeln:berg Corporation, 15200 Foltz grinding from the sollid. eliminating the need tor pre- IndustriallParkway, Cl'eveland, OH 44136. Or, phone liminarY mining of worms, threads and flotors ..tt'stwo (216)572~.21'OOfor an extra FAST response. machines in one ... with the, high degr,ee of accuracy you'd expect from Klingelnberg'. @KLIING'E!LNBIER'G INVOLUfOMETRY ••• (continued from page 42) r-rI Eq.U Eq.14 ment of a gear of infinite size, but with teeth the size of the gear 'cut parallel Itothe gear axis. By inclining the cutter at a dilferent we want to cut. As a result, the "involate" sides of the teeth are angle it can cut a helical gear. The cutting activity closely essentially straight lines. To generate a tooth, the basic rack resembles a worm. gear drive. Hobbing is the most popular and moves in a. straight line like a. rack and pinion arrangement, fastest method for cutting gears. with the cuttingaction being up and down axially to the gear. In our calculations it is necessary to determlnethe position In reality, instead of having a very long rack, segments are ar- of the basic rack with the gear being generated in order to ranged around a cylinder and offset to create a helix. With the establish the form diameter and root radius and determine axis of the cutter inclined, the teeth will contact the gear being (continued on page 48) ASM Conference on NEAR NE,T SHAPE ,MANUFACfURIN'G 8-10 Novctnber 1988 - ColUIII.bus,Ohio ASH\. Spc}IIsoreiJl7y tbe ,\lateRals Shllping Technology Oil/lsl'on of.ASM INTERNATIONAl. Co-Sponsored by ... The Enginee.ring Resetu'ch Center for Net Sbape Manufacturing (Obio tale, 1jlvel"5ily) 'c:::1 Forging Industry Association .' tnuestment Caslirlg lrlSlil!m~ • The Meialltlrgtccil Soclel)' IF YOU OR YOUR COMPANY ARE INVOLVED WITH ••. POWDER M£fALS FORGING ,CASTINGS EXTRUSION a DRAWING SHEET FORMING MACIUNING • Injection • Process Design • Iodellng '. Roll Drawing • Comparison of • MnS Additions • IHIIP • c!,o .' Directional .' Uhrafine Wire Models .' flnil Elemen; • Spr.l)' Deposit ion. • Ring Rolling Solidification '.' Cold Impact Extrusion .' pring Back Design • IPress & inter '. Precision FOIging • Squeeze Casting .' Process Analysis • PM Pan .' lubrication • Precision Ca:;!Lng Method • Fin.ile Element ••• YOU NEED ro ATTEND THIS CONFERENCE H Sessions .on.... fORGINGS· • MACIIiNING &: FINISHING' SHEET FORMJNG° POWDER METALWRGY "eXTRUSION &:DRAWING • CASTINGS pJu~ .a .tour olOhiQ Stille University's ENGINEE!l'NG .R..liSMHCH CENTER FOR NET SHAPE MA~UFACTVRlNG RcglStr.ltion Fee are:, S32S for Members of ASM, TMS, IHA. I I and ERCfNSM; 5370 for Non Members. for more Information. conlaC:t ASM [ TERNATIO AL's Member~ustomer Service Center at 216/338.-5151, ext. 703; fax IlUJ"ILDer 2Hi/j,S6j4; ------or dip the form below and ------mail t.o: MEMBER/'C' STOMER SERVICE ------. CE TER- ASM INTERNA.1'IO ------11AL"' Metals PJrk, au 4 0-3 o I am rnterested In mending, the ASI!!!'CONfERENCE ON IIIMR NIT SHAPE MANllFACTL'RI G, Send me the complete conference program. o tcannot attend '1he A_M CONfERENCE 0 NEAR ET SHAPE MAN' rAC· TURING. bUI I would like 10 order prnce,ed;ngs of the Conference, Send me price mformatlon on lhls publication, "'DDRESS; _ PI~ send me a FR.EEmembership inform~Uonpacket on ASM fN .ER A- o crrv, STATE -"tI'lI'OST ...L COI)£, _ TIONAL I am tnrerested in learning more about -A.Mi'" Technical Produet.s and Services in the field of Engino:ted Malerlal3, ami how these pru N ...1Mf. TELF.PfI[)NP.. f...x Nl' !DEll CIRCLE A-39 ON READER IREPLY CARD September/octobef 1988 45 ----- (~I.o~~SSII~II~D COMPUTER AIDS GEAR' ESTIMlATiNG GEAR CALCULATI'O,N PROGRAMS F'OIR T:HE The COSTIMATOR® computer GEA.IR MAINUFACTUIRIE.:R , aided cost estimating system insures ~speed and consistency in the difficult liNEXPIENSIVE,. EASY TO RUN GEAR DESIGN/ANALYSISfor IBM PC s , task of estimating the costs of all ON IBM COMPATIBLES compatibles.GEARCALC($995): design op- types of gears. timum spur & helical gearsets from appli· Used by small shops and Fortune cation data. AGMA218 ($1495): calculate 500 companies through1out the EXl'EAN'AL liNCH ID:P) pitting and Ilendingfatiguelives per AGMA country .. INTERNAL METRIC ,(Mod.) Std .: 218.01. SCORING +- ($495): S'PU'R analyzefplot scoring & wear pro- IHIEUCAL babilities. Fast, menu-driven pro- For complete information STRAIGHT a.eViEL-(104) grams work as one system (eembln- contact us today. SPIRAL BEVEL. 1#16& 27 GRD BLANK AND SUMMAFUES ed price $.2495), do extensive error- checking. Comprehensive manuals IManulacturers t:echno'iogies, tne .. For Brochure and More Information CAU OR WRIITE (theory, operation, examp'les) in· 59(3, lnterstateDr, UNIVERSAL GEAR' CO. INC .. eluded. Demo avail'abl'e. GEAR.TECH West Spring;field, MA 01089 - P.O.Box A.G., Anza, CA. 92306 - . Software, Inc., 1017 Pomona A.ve., (41,3) 733·1972 (I14) 763-4616 .Albany, CA 94706 (415) 524·0668. CIRClE A-16 ON READER REPLY CARD CIRCLE A-117 ON IREADER !REPlY CARD cn~CUE A·] 8 ON READER R.EP,lY CARD COMPUTER AIDS 1- SERVICE GEARS SPLINES AiPiPlICAT;I,ON GEAR: TES,TIING AND TESTING:-iDEVELOPMENT DESIGN &. TOOLIING DIESI:G'N FACIUTIES CONSULTING & SOFTWARE Total Gear TeChnology from comput,er ,aided • Let us design your gears for greater • GEAR DESIGN (NOiSe: - STRENGTH) strength. engineering of design to final • ROTATING GEAR (TORQUE - SPEED CONTROL) TEST MACHINes. • Let us install our software and train ,application testing . .' SINGLE TOOTH BENDING IFATIGUE you r people in the latest g.earand cut- • Loaded Contact and Axle Deflection TESUNG. ter computing methods ..Trialiperiods • Fatigue Strength and Durability , • STATIS-';ICAl PlJ>iNNING - ANALYSIS. available. • Constant Velocity Testing . '. WROUGHT STEELS, SINTEREO • You can benefitfrom our many years • Instrumented Vehicle Sound Analysis I METALS, NON-METALLIC MAT'L.S. of experience wi,th computers, gear • G-Age with Zeiss CMM '. CAD FACILITIES FOR LOW COST design and manufacturing. • Single Hank SET-UP. • Softwar·e for IBM and compatible I. ,Finite Element Analysis '. CUSTOM TEST MACHIINE IDESIGN computers. • IEXPERIENCEDPEASONNEL VAN GEIIJlEN~·REECE ENGINEERING 'The 'Gleaso:n Works 1502 Grand Blvd .. 1000 University Ave. PACKER ENGINE:ERING Cedar Falls, I'owa 50613 Rochester, New York 14692 312/355-5722, ext. 21'4 (319') 266-4674 (716) 461-8026, Larry Palmateer BOX 353, NAPERVILLE, IL 60566 --' CIRCLE A-32 ON READER REPLY CARD CIRCLE A~2 ON READER REPLY CARD CiRClE A-36 ON READER ,REPLYCARD, Bring in new customers for your business by advertising in GEAR TECHNOLOGY, The Journal of Gear M,anufacturing. Can (312)437-6604 Rates: Classified ,Display-per inch (rnlnl- Payment.: Full payment must accompany lMater:ia.ls Deadline: Ads must be recei.ved mum 3") 1.X·$120, lX·$110, 6X·$100. Type classified ads. Mail copy to Gear Technology, by the 25th of the month, two, months prior will be set to advertiser's layout or Gear P.O. Box 1426, Elk Grove Village. It 60007. to publication. Acceptance: Publisher Technology will set type at no extra charge. Agency Commission: No agency commission reserves the right to accept or reject Word Count: 35 characters per line, 7 lines on elassifiacs. classified advertisements at his discretion. per inch. 46 GearTechnology HELP WANTED SUBCONTRACT M~~UFACTURING MANAGE:R GEAR IPERS'ONNEiL GEAR GRlND.ING NIE,W AIRCRAFT' FACIIUTV ENGINEERS rOOTH' An aggressive, aircraft & precision parts GEA'R GRINDER.S & HONING .oNLY MACHINISTS, manufacturer (gearboxes" gears, shafts, Produclliol1 Quantities MANAGEMENT/SUPERVISIONI housings) la seeking an experienced 314" P.O. to 27.S" P.lD. Manufacturing Mana-ger withlhe ability I PRODUCTION CONTROL to direct the msnufactu ningoperations & QUALlTV OONTROLIINSPECTION 3.5 D.P'. and 11 .. Face growth of an ,ex:paJ1c1lng200 person :SALIES 'Gear Tooth IF,inishingl faclHty. The candidate we seek must ACR has moved 10 a new 85,000 square lis our ,OnJy Business Ihave working knowl'edge of aircraft foot fa.cility in Mt. CI'emens, Michigan. As pariS" q,uaJity requirements" tools, fix- have notumfng. hobbing or a result, the above and other new pOsitions, we tures" proceSsing & manufacturing. lin shaping capability return we ,off,eran attractive salary, e.x- 8J'e beinQl created. If you are in the the gear business in any capacity, are ambitious ' cellent benefits & a.futur,e dir,ected en- and thinking of making a. move, send us ALIlE.GHIENIY ,GEAR C'OR;P. viro.nment.IP,lease submit your resume your resume. withsal'ary history to: 23 Dick IRoad ...... ACR INDUSTRIES. INC. ACR INDUSTRIES, INC., loe:,e.~.'~ 15375 Twenty Three Mile Ad. 15375 Twenty-Three Mile Ad •. ¢. FAX (716) ~i711 f Mt. Clemens, M148044-9680 M•. Clemens, MI 48Q44-9680 :'"lVJ)l CU~CLE A-23 .oNI READE'R REPiLVCARD SALES AND SERVICE REBUILDtNG A FULL SERVICE COMPANY SUPPLYING THE AEROSPACE H:O'BB'E'R IREBUILDIING AND COMMERCIAL mDUSTRIES WITH PRECISION PRODUCTS. SPECIALlST:S Having! trouble meeting today's d& mand quality col1trol to'lerances? V1NOQ®B-lRADIUS/ANGLE WHEEL DRESSER iLet our factory trainedl and ex- VlNCQ® SPLINEDICATORS AND WEAR GAGES perienced staff return your VINCQ® GEAR MEASURING INSTRUMENTS machine to optimum operating GEAR COMPONENTS AND PARTS oondition. .oPERATIONAL GEAR GRINDING We spectattae iln repairingl, SPUNE GAGES AND ARBORS rebuilding aM mode:miztng aU MASTER GEARS makes o. 'hob'bers. - • Cleveland Rigjdhobbers • Gould & Ebe~ha.rdt TIFCO SPLINE, INC. • Barber Co'lmall VINOO® PRODUCTS PRESSftlATION IINC. 2900S,ANTHONY DRIVE WIXOM. MICHIGAN 48096 522 Cottage Grove Road (313) 624-790.2 Bloomfield, Conn. 06002 FAX (313) 624-1260 (203) 242-8525, CIRCU 1'.-3,11,ON READER REP,lY CARD CIRCLE 1'.-41 ON READER REPLY CARD There's stillltime .... Tell our advertisers order dosing date for a olasshied ad in the you saw it in !Nov.lDee. issue lis Seipt. 110th. GEAR TEICHNOtiOGY September/Octob9ll' 1988 47 aJSlNESS OPPORTUNITY It's true. our Consumer Information Catalog is tilled with booklets that _ As. can answer the questions American ••• FOR G!EAR CONSULTANTS Amencan consumers ask most. Put EXCIiTING NEW GEAR OIAG- TOsatisfy every appetne, the NOSTIC TECHNOLOGY to work Consumer Information Center puis solving your clients' most criticallgear together this helpful Catalog problems. quarterly containing more than 200 federal pubhcancns you can order. ,Fortune 500s are nowgelfing BREAK- trs free, and soare almost half of Ihe THIROUGH iENHANCEMENTS IN booklets it lists. Subjects like MONIITORING GiEAR PERFOR- nutrition, money management, MANCE AND CONDITION .... for health and iederal benefits help you make the right choices and end of Iline (QIA), predictive decisions. . maintenanoe& engineering and testing applications. - So get a slice or American ·J'oin MTC jin applying this technology .. opportunity. Write today for your free Catalog: . CaJl1EO PAGE for more information. IMONITORlNG JEOKNOLOG,Y 'CORPORA110N Consumer Infor::-mation center 2n9 Han\and Roa.d Department AP Falls Church, VA 22043 Pueblo. COlorado, 81009 Call (703), 16198-5520 US General Sep'!h.l'~ AdmIMLSlr,atIOr\ CIRCLE A-40' ON READER REPLYCARD INVOLUTOMETRY. ,.. (continued from page 45) ,anI = Normal pressure angle at RI whether 'the cutter will interfere with the outside diameter ,of the al == Transverse pressure angle at Rl gear .. Usually a gear tooth thickness is given, and the Rl = Given pitch radius corresponding position of the basic rack is computed. R, - Base radius Given the tooth dimensions in the plane of rotation of a T 1 = Given tooth thickness at Rl helical gear, to determine the position ofa mating rack of di!- X == Dist, gear center to hob tip fe.lentcircular pitch and pressure angle (same base pitch), use Pn2 = Normal pitch of rack the fonowing formulae. Base pitch must be equal (Pnl • Cos al = Pn2 .. C.oSa2) Sin (jb = Sin {31 . Cos cxnl The position of the rack formula results in. at figure for the root radius of the gear. The following formula wi]] give the minimum root radius without undercut. By comparison, it can ".·-t:l _ Sin (jb _ Sin {31. Cos anI Sm"'2 - - be determined if undel1cutting is taking place. Undercutting is COS CXn2 COS an2 undesirable. By 'taking the minumum root radius without undercut a new tooth thickness for the gear can be determined. 'T Tan O'n2 Ian 0'2 = .------'= R2----- R, By keeping one gear from undercutting, the mating gear should Cos 13'2 Cos 0'2. be checked to determine if it may be undercut as a result of shifting t.ooth thickness. = R2 - a. X + _--",,,....1 _ Giventhedimension.s ·ofa hob, the helix angle and number 2 Tan 0'.2 of teeth in a helical gea.r, to determine the mlnlmum root radius to avoid undercut, use the following formulae, '. T . '/!'·R .2 ..R2 '__ 1_ + inv a1 - inv a2 - (16) Z Tan ahb ----=; R=---- IIII2 ·R1 Z 2· Dp· Cos 13 Tan 0' = Cos fJ where Run.: = R· Cos2a - r : (1 - Sin a) (17) 131 = Given helix angle at Rl where fJb= Base helix angle an2 = Pr-essure angle of rack Z = Number of teeth in gear (1'2 = Transverse pressure ang1e of rack an" = Pressure ang1e of hob R2. = Pitch radius with rack 0' = Pressure angle in plane of rotation a = Addendum of rack fJ = Helix angle Z = Number of teeth DP = Diametral pitch of hob Pnl = Normal pitch at Rl r = Radius of hob tooth tip 132 == Helix angle for mating rack Rune = Minimum root radius wi 0 undercut .1 48 Gear Technology In Seven Decades, the~Baste rrit:'lciple~ of GeC!rMaking Havent Changed ...Untd Now Introducing PHOENIX INO CRADI!.iE., NO' ECCENTRIC. High Illepeatablllty. ..•sec~p~to- NO SWIVEL NO CUT1iERTlI% letup and'imathlne--to"'machine PHOENIX is a revolutionary Tlhe variables disappear. Setup in- new gear machine thareumlnates consistencies between batches, or all of the mecnankal setup adjust- between machines, are replaced ments necessary with convenuonal with CNC control of all setup machines used for cutting or grind'- parameters. rrial-and",errm adjust- ing bevell and hypoid gears~ The ments are eliminated, making scrap result Both uptime and flexibHity a real rarit)~. are multiplied. ,Ceslgnedf:orautomatJon .• " Only 6 axes, of,lmIchlne motion I now or later 'create aU ,gear ,toot.hl geometry The PHOENIX design, combined Wrth PHO,ENIX 6 CNC~control- with its high repeatability and com- led axes, do it all. The secret is PHOENIIX p~ehensjve CNC control make it to control these 6 simple to automate part and tool Gleason's ability SlX'Iludsl precision ellmlnat,es uadl'e, axes so precisely that [he mechan- eeeentrlc, Clutter tilt •.SWlvell .... handling. eel/and system integra,- ical motions we aJi know are no ,ilndtradlUonaJ setllp. don has never been easier. longer required. Cetans at IIMTS, '88 Changeove,r Find out more In mfnutes, about how the most Random batch dramatic improve- processing can be as ment in ,gear produc- simple as calling up [ion in decades can a program and put- work into your plans. ting the tooling in See us .at Booth pla-ce. Cuner length #I 6301 or contact us adjustments are ac- now. The Gleason complished auto- Works, Division of matically Training Gleason Corporation. requirements are P.O. Box 22970, minimized. Roches[e~NY 14692 j7161473-1000. e,IMTS Gleason :IWJIIIIIMIUIl'.aII1I11l" The! W'arlcllaf Giellr,lng BOOTH630'l CIRCIJE ."·2.0 ON IR,EADER IREPlV CA_RD HURTH Rotary Deburrlng Machines and TOols fOr fast, reliable gear chamfering and deburrlng • Short cycle nmes • High ouallty • LOwcost per Piece Gearsand cluster shafts of up to 4 gears of different configurations are chamfered and deburred simul- taneously on different versions of the HURTHZEAline. HURTHloading and unloading equip- ment can be fully integrated in all ZEAdeburring automatics. Also, HURTHbrings you the right tool system for every application. From the basic version, consisting of two deburring tools and a meshing gear, right up to tools incorporating master gears, side trimming discs and tools, burnishing wheels and pressure plates, HURTHhas the right tool for your application. We would be pleased to review your gear deburring and chamfering requirements and invite your lncurv. • Rotary deburring tools on a four station unit machining gears for second and fourth. while two side trimming tools and two TiN side trimming discs remove secondary burrs. • Automatic rotary deburring machine. SEND FOR FREE BOOKLET - HURTH Standard HN 115 206 -a practical guide for //lJiiij'" selecting the appropriate chamfer. 7~ARCUT SALES, INC. &/~~~61 Industrial Park Drive Farmington Hills, MI48024 313/474-8200 Telex 230-411 STARCUT SUPPUED PRODUCTS AND SERVICES star MachilDe Tools I star cutting Tools / Gol'dstar coatl'ng!s / Hurt,h Machine TOols I Mikron IHobblng! Machines and' Tools /TiN Coating Systems /Stieber Cil,ampln,g TOOlS CIRCLE A~26 ON READER REPLYCARD